Hsiang-Ping Huang math1060fall2008-1 - math.utah.edu · Usually you can attempt a problem as many...

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Intro to WeBWorK is due : 09/01/2008 at 12:00pm MDT.The

webpagefor the course contains the syllabus, grading policy and other information.

This homework set is to introduce you to WeBWorK.

The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are makingsome kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you arehaving trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor forhelp. Don’t spend a lot of time guessing – it’s not very efficient or effective.

Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,you can if you wish enter elementary expressions such as 2∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e∧ (ln(2)) instead of 2,(2+ tan(3))∗ (4− sin(5))∧6−7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.

You can use the Feedback button on each problem page to send e-mail to the professors.

1. (3 pts) 1010Library/set1 WebWork Demo/m1p165.pgEnter here the expression 1

a + 1b .

Enter here the expression 1a+b .

Correct Answers:

• 1/a+1/b• 1/(a+b)

2. (3 pts) 1010Library/set1 WebWork Demo/s1.p29.pg

For each of the WeBWorK phrases below enter a T (true)if the two given phrases describe the same algebraic expressionand a F (false) otherwise. One way you can decide whether thephrases are equivalent is to substitute specific values for a, b,etc. If you get two different results the two phrases are certainlynot equivalent. If you get the same values there is small chancethis happened accidentally for just that choice of particular val-ues. In any case, pay close attention to when these phrases areequivalent and when they are not, it will help you tremendouslywith future WeBWorK assignments.

a+b2 (a+b)2

a2 +b2 (a+b)2

a∗b∗ c a∗ (b∗ c)

a/b/c a/(b/c)

Correct Answers:

• F• F• T• F

3. (3 pts) 1010Library/set1 WebWork Demo/s1p1.pgThis first question is just an exercise in entering answers intoWeBWorK. It also gives you an opportunity to experiment withentering different arithmetic and algebraic expressions intoWeBWorK and seeing what WeBWorK really thinks you are do-ing (as opposed to what you believe it should think).

Notice the buttons on this page and try them out before movingto the next problem. Use the ”Back” Button on your browser toget back here when needed.

”Prob. List” gets you back to the list of all problems in this set.

”Next” gets you to the next question in this set.

”Submit Answer” submits your answer as you might expect, butthere may be other ways to do so. Specifically, in this problem,there is only one question. In that case you can submit youranswer by typing it into the answer window and then pressing”Return” (or ”Enter”) on your keyboard. But even in this case,you can also type the answer and click on the ”submit” button.There is no harm in submitting an answer even if you are notquite sure that it’s correct, since if it is not you have an unlim-ited number of additional tries. On the other hand, it is usuallymore efficient to print your own private problems set, work outthe answers in a quiet environment like your home, and then sitdown in front of a computer and enter your answers. If someare wrong you can try to fix them right at the computer, or youmay want to go back and work on them quietly elsewhere beforereturning to the computer.

Pressing on the ”Preview Answer” Button makes WeBWorKdisplay what it thinks you entered in the answer window. After

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using ”Preview” you can modify your answer and use a ”Pre-view Again” button.

”images” denotes the ordinary display mode on your worksta-tion.

”Logout” terminates this WeBWorK session for you. You canof course log back in and continue.

”Feedback” enables you to send a message to your instructor,and the WeBWorK assistants. If you use this way of sendinge-mail the recipients receive information about your WeBWorKstate, in addition to your actual message.

The ”Help” Button transports you to an official WeBWorK helppage that has a more information than this first problem.

”Problem Sets” transports you back to the page where you canselect a certain problem set. When you do this particular prob-lem in this first set, there is only one set, but eventually therewill be 13 of them.

For all problems in this course you will be able to see the An-swers to the problems after the due date . Go to a problem,click on ”show correct answers”, and then click on ”submit an-swer”. You can also download and print a hard copy with theanswers showing. These answers are the precise strings againstwhich WeBWorK compares your answer. If the answer is analgebraic expression your answer needs to be equivalent to theWeBWorK answer, but it may be in a different form. For ex-ample if WeBWorK thinks the answer is 2 ∗ a, it is OK for youto type a + a instead. If WeBWorK expects a numerical an-swer then you can usually enter it as an arithmetic expression(like 1/7 instead of .142857), and usually WeBWorK will ex-pect your answer to be within one tenth of one percent of whatit thinks the answer is.

Most of the problems (including this one) in this course willalso have solutions attached that you can see after the due dateby clicking on ”show solutions” followed by ”submit answers”.The solutions are text typed by your instructor that gives moreinformation than the ”answers”, and in particular often explainshow the answers can be obtained.

Now for the meat of this problem. Notice that the answer win-dow is extra large so you can try the things suggested above.

Type the number 3 here:.

Try entering other expressions and use the preview buttonto see what WeBWorK thinks you entered. Return to this

problem to try out things when you get stuck somewhereelse.

Here are some good examples to try. Check them all out usingthe Preview button. (In later questions on this set you will getto use what you learn here.) Never mind that you may have al-ready answered the correct answer 3. Once you get credit for ananswer it won’t be taken away by trying other answers.

a/2b versus a/2/b versus a/(2b)

a/b+c versus a/(b+c)

a+b**2 versus (a+b)**2

sqrt a+b versus sqrt(a+b)

4/3 pi r**2 versus (4/3) pi r**2 (In other words, if you are notsure use parentheses freely.)

Note: WeBWorK will not usually let you enter algebraic ex-pressions when the answer is a number, and it will only let youuse certain variables when the answer is in fact an algebraic ex-pression. So the above window, and the opportunity for exper-imentation that it offers is unique. Make good use of it!

Presumably this has been your first encounter with WeBWorK.Come back here to try things out and to refresh your memory ifyou get stuck somewhere down the line.

Correct Answers:

• 3

4. (3 pts) 1010Library/set1 WebWork Demo/s1p2.pgThe purpose of this exercise is to illustrate further the use of thebuttons on this page and to show you the most common way inwhich WeBWorK processes partially correct problems. Try en-tering incorrect answers in the answer fields below, to see whathappens. (This time WeBWorK will reject algebraic expressionssince I told it to expect a numerical answer.)

Type the number 4 here: .

Type the number 5 here: .

Correct Answers:

• 4• 5

5. (3 pts) 1010Library/set1 WebWork Demo/s1p3.pg

In the first few problems, now that you are familiar with thebasic mechanics of WeBWorK, you will be asked to evaluatesome arithmetic expressions and enter the answer as a numberinto WeBWorK. You may of course use a calculator. In laterproblems you will be able to enter the answer as an arithmetic

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expression, but at present your answer must be a number suchas 4, -4, or 17.5.

Evaluate the expression7(8 + 5) = . (Remember that by conventiona missing arithmetic operator means multiplication .)

Correct Answers:

• 91


Evaluate the expression9(7−8) = .

Correct Answers:

• -9


Evaluate the expression8/(2+2) = .Enter your answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Correct Answers:

• 2


Evaluate the expression11− (5−2) = .

Correct Answers:

• 8


Evaluate the expression5− (6−8) = .

Correct Answers:

• 7

10. (3 pts) 1010Library/set1 WebWork Demo/s1p8.pgThis problem illustrates the standard rules of arithmetic prece-dence:

Multiplication and Division precede Subtraction and Addition.Among operations with the same level of precedence, evalua-tion proceeds from left to right.However, expressions in parentheses are evaluated first.

Evaluate the expression6×3−2×5 =

Evaluate the expression6× (3−2)×5 =

Evaluate the expression6× (3−2×5) =

Correct Answers:

• 8• 30• -42

11. (3 pts) 1010Library/set1 WebWork Demo/s1p9.pgThis problem provides more illustrations of the use of parenthe-ses.

Evaluate the expression9−10−5−4 =

Evaluate the expression9− (10−5)−4 =

Evaluate the expression9− (10−5−4) =

Evaluate the expression9− (10− (5−4)) =

Correct Answers:

• -10• 0• 8• 0

12. (3 pts) 1010Library/set1 WebWork Demo/s1p10.pgThe key idea in Algebra is to use variables in addition to num-bers. Sometimes we need to replace variables with specificnumbers. That’s called evaluating an algebraic expression.For example, if a = 2 then 3a = 6, and we say that we evaluatedthe expression 3a at a = 2. We’ll do this sort of thing all semes-ter long, and in this problem you get your first experience withevaluating algebraic expressions. Again, the emphasis in theseexercises is on understanding the rules of arithmetic precedence.

Let a = 7, b = 4, c = 13.Then a−b/c =and (a−b)/c =As usual, enter your answers as decimal numbers with at

least 4 digits.Correct Answers:

• 6.69230769230769• 0.230769230769231

3

13. (3 pts) 1010Library/set1 WebWork Demo/s1p11.pgLet r = 5.

Then 4/π∗ r =and 4/(π∗ r) =Correct Answers:

• 6.36619772367581• 0.254647908947033

14. (3 pts) 1010Library/set1 WebWork Demo/s1p12.pgLet a = 3, b = 4, c = 6, d = 8.

Then a−b/c−d = ,

(a−b)/(c−d) = ,a− (b/c−d) = , anda−b/(c−d) = .

Correct Answers:• -5.66666666666667• 0.5• 10.3333333333333• 5

15. (3 pts) 1010Library/set1 WebWork Demo/s1p13.pgThe next three problems are like the preceding three, except thatyou need to get all answers correct before WeBWorK will giveyou credit. This will be true for many problems in this class.The purpose of insisting on all answers being correct is to en-courage you to think about the whole context of the problemrather than the individual pieces.

Let a = 2.5, b = 3.5, c = 5.7.Then a−b/c =and (a−b)/c =Correct Answers:

• 1.8859649122807• -0.175438596491228

16. (3 pts) 1010Library/set1 WebWork Demo/s1p14.pgLet r = 7.9.

Then 4/π∗ r =and 4/(π∗ r) =Correct Answers:

• 10.0585924034078• 0.161169562624704

17. (3 pts) 1010Library/set1 WebWork Demo/s1p15.pgLet a = 2.1, b = 4.1, c = 7.1, d = 8.7.

Thena−b/c−d = ,

(a−b)/(c−d) = ,a− (b/c−d) = , anda−b/(c−d) = .

Correct Answers:• -7.17746478873239• 1.25• 10.2225352112676• 4.6625

18. (3 pts) 1010Library/set1 WebWork Demo/s1p16.pgIn this and the following problems you will practice en-tering algebraic expressions into WeBWorK. Remember theRules of Arithmetic Precedence and use parentheses to makeyour meaning clear. Most of the difficulties students have withWeBWorK are due to not appreciating the precise rules that gov-ern the interpretation of what you enter. This is not just a matterof WeBWorK understanding what you are trying to say, Therules are used universally all over the world. Appreciating andapplying them properly is also crucial, for example, in computerprogramming. Make sure you understand what’s going in theseproblems. If you enter a wrong expression use the Preview But-ton to see what WeBWorK thinks you have entered.

We start simply. Enter here the expression a+b.Correct Answers:

• a+b

19. (3 pts) 1010Library/set1 WebWork Demo/s1p17.pgEnter here the expression

a+12+b

Enter here the expressiona+bc+d

If WeBWorK rejects your answer use the preview button tosee what it thinks you are trying to tell it.

Correct Answers:• (a+1)/(2+b)• (a+b)/(c+d)


11a + 1

b

Enter here the expressiona+b+11+ 1

a+b

Correct Answers:• 1/(1/a+1/b)• (a+b+1)/(1+1/(a+b))


Enter here the expressionab + c

def + g

h.

Correct Answers:• (a/b+c/d)/(e/f+g/h)

4


The square x2 of a number x simply means the product of xwith itself. For example, 32 = 3∗3 = 9. You can enter a numbersuch as 32 as 3**2 . (An expression such as 32 or x2 is calleda power . We will learn a great deal more about powers duringthis semester.)

Enter here the expression x2

The square root√

x of a number x is a number whose squareequals x. For example

√25 = 5 since 52 = 5∗5 = 25.

To enter square roots you can use the function sqrt . Forexample, to enter the square root of 2 you can type sqrt (2).

Enter here the expression√

aCorrect Answers:

• x**2• sqrt{a}


√a+b

Enter here the expressiona√

a+bEnter here the expression

a+b√a+b

Correct Answers:• sqrt(a+b)• a/sqrt(a+b)• (a+b)/sqrt(a+b)


Enter here the expression√x2 + y2

Enter here the expression

x√

x2 + y2

Enter here the expressionx+ y√x2 + y2

Correct Answers:• sqrt(x**2+y**2)• x*sqrt(x**2+y**2)• (x+y)/sqrt(x**2+y**2)


Enter here the expression

−b+√

b2−4ac2a

Note: this is an expression that gives the solution of a quadraticequation by the quadratic formula . We will learn much moreabout it later in the semester.

Correct Answers:

• (-b+sqrt(b**2-4*a*c))/(2a)

26. (3 pts) 1010Library/set1 WebWork Demo/s1p24.pgConsider the following expressions:

A = a+bc

and

B =a+b

c

For each of the WeBWorK phrases below write A if they de-fine A and B if they define B.

You need to get all answers correct before obtaining credit.

a+b/c (This is the standard way to enter A, so enter A).

(a+b)/c (This is the standard way to enter B, so enter B).

((a+b)/c)

a+(b/c)

(a+(b/c))

Correct Answers:

• A• B• B• A• A

5

27. (3 pts) 1010Library/set1 WebWork Demo/s1p25.pgConsider again the formula for the solution of a quadratic equa-tion:

x =−b+

√b2−4ac

2aFor each of the WeBWorK phrases below enter a T (true) if

the phrase describes x, correctly, and a F (false) otherwise.

You need to get all answers correct before obtaining credit.

-b+sqrt(b**2-4*a*c)/2a

(-b+sqrt(b**2-4*a*c))/2a

(-b+sqrt(b**2-4*a*c))/(2a)

Correct Answers:

• F• F• T

28. (3 pts) 1010Library/set1 WebWork Demo/s1p26.pgMore of the same.

(-b+sqrt(b**2-(4*a*c)))/(2a)

(-b+(sqrt(b**2-4*a*c)))/(2a)

((-b+sqrt(b**2-4*a*c))/(2a))

(-b+(sqrt(b**2-4*a*c))/(2a))

(-b+sqrt(b*b-(4*a*c)))/(2a)

Correct Answers:

• T• T• T• F• T


For each of the WeBWorK phrases below enter a T (true) ifthe two given phrases describe the same algebraic expressionand an F (false) otherwise. One way you can decide whether thephrases are equivalent is to substitute specific values for a, b,etc. If you get two different results the two phrases are certainlynot equivalent. If you get the same values there is small chancethis happened accidentally for just that choice of particular val-ues. In any case, pay close attention to when these phrases areequivalent and when they are not, it will help you tremendouslywith future WeBWorK assignments.

a+b b+a

a+b+ c a+(b+ c)

a−b− c a− (b− c)

Correct Answers:• T• T• F


More of the same.

a+b2 (a+b)2

a2 +b2 (a+b)2

a∗b∗ c a∗ (b∗ c)

a/b/c a/(b/c)

Correct Answers:• F• F• T• F


In mathematics, lower and upper case letters mean differentthings. The letter a is not the same as the letter A. Keep that inmind when answering the questions below, using T or F as inthe preceding questions.

a a

a A

a+A A+a

Correct Answers:• T• F• T


Much of this course will center around the manipulation ofalgebraic expressions, often with the goal of solving an equa-tion. This exercise is the first step in this direction. Again, indi-cated with T or F if the two expressions are equivalent.

6

a∗ (b+ c) a∗b+a∗ c

1/(a+b) 1/a+1/b

1/a/a 1/(a∗a)

Correct Answers:• T• F• T


The reason why Mathematics is required for so many sub-jects is that it can be used to solve problems outside of mathe-matics, the dreaded word problems . There will be many wordproblems in this class, usually leading to a mathematical prob-lem of the kind we are discussing at the time. Students don’t

like word problems because they involve the extra layer of con-verting the word problem to a math problem. But keep in mindthat math classes are the only kind of classes you take wheresome problems are not word problems!

This first word problem of this course can be solved by derivingand solving an equation, but it can also be solved essentially byguessing and modifying the answer until it fits, without any al-gebraic manipulation. We will revisit it in the future in a morecomplicated setting.

You buy a pot and its lid for a total of $ 11. The sales per-son tells you that the pot by itself costs $ 10 more than the lid.The price of the pot is $ and the price of the lid is $

.Correct Answers:

• 10.5• 0.5

Generated by the WeBWorK system c©WeBWorK Team, Department of Mathematics, University of Rochester

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Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Functions is due : 09/08/2008 at 03:00pm MDT.The


This assignment deals with functions. The relevant material can be found in sections 1.4-1.6 of the text.




1. (5 pts) Library/ma112DB/set6/sw4 1 33.pgGiven the function f (x) = 5−7x + 13x2. Calculate the follow-ing values:f (a) =f (a+h) =f (a+h)− f (a)

h = for h 6= 0

Correct Answers:• 5 - 7*a + 13*a**2• 5 - 7*(a+h) + 13*(a+h)**2• -7+2*13*a+13*h

2. (5 pts) Library/ma112DB/set6/sw4 1 31.pgGiven the function f (x) = 16. Calculate the following values:f (a) =f (a+h) =f (a+h)− f (a)

h = for h 6= 0

Correct Answers:• 16• 16• 0

3. (5 pts) Library/ma112DB/set6/sw4 1 19.pgGiven the function f (x) = 5|x−4|. Calculate the following val-ues:f (0) =f (2) =f (−2) =f (x+1) =f (x2 +2) =In your answer, use abs(g(x)) for |g(x)|!!!


• 5*abs(x+1-4)• 5*abs(x**2+2-4)

4. (5 pts) Library/Rochester/setDiscrete4Functions/ur dis 4 1.pg

Determine whether f is a function from Z to R.Enter ”Y” for yes and ”N” for no.

1. f (n) =±n2. f (n) = 1

n2−163. f (n) = 1/(n2 +1)4. f (n) =

√n2 +4

Correct Answers:• N• N• Y• Y

5. (5 pts) Library/Rochester/setDiscrete4Functions/ur dis 4 3.pg

Determine if each of the following functions from {a,b,c,d}to itself is one-to-one and/or onto.Check ALL correct answers.(a) f (a) = b, f (b) = b, f (c) = d, f (d) = c

• A. onto.• B. neither one-to-one nor onto.• C. one-to-one.

f (a) = d, f (b) = a, f (c) = c, f (d) = b• A. neither one-to-one nor onto.• B. onto.• C. one-to-one.

f (a) = c, f (b) = d, f (c) = a• A. onto.• B. one-to-one.• C. neither one-to-one nor onto.

Correct Answers:• B

1

• BC• B

6. (5 pts) Library/Rochester/setAlgebra15Functions/sw4 1 11.pgGiven the function f (x) = 3x2−2x+1. Calculate the followingvalues:f (−2) =f (−1) =f (0) =f (1) =f (2) =

Correct Answers:

• 17• 6• 1• 2• 9

7. (5 pts) Library/maCalcDB/setAlgebra15Functions/srw2 1 44.pgThe domain of the function

4x2 +17

isWrite the answer in interval notation.Note: If the answer includes more than one interval write theintervals separated by the union symbol, U. If needed enter −∞

as - infinity and ∞ as infinity .Correct Answers:

• (-infinity,infinity)

8. (5 pts) Library/maCalcDB/setAlgebra15Functions/p6.pgFind domain and range of the function

√x+11

Domain:Range:Write the answer in interval notation.Note: If the answer includes more than one interval write theintervals separated by the union symbol, U. If needed enter −∞


• [0,infinity)• [11,infinity)

9. (5 pts) Library/maCalcDB/setAlgebra15Functions/srw2 1 3.pgExpress the rule ”Subtract 18, then square” as the function

f (x) = .

Correct Answers:

• (x-18)*(x-18)

10. (5 pts) Library/maCalcDB/setAlgebra15Functions/srw2 1 1.pgExpress the rule ”Multiply by 7, then add 19” as the function

f (x) = .

Correct Answers:• 7*x+19

11. (5 pts) Library/maCalcDB/setAlgebra15Functions/srw2 1 53.pgThe domain of the function

3√t−74

is .Write the answer in interval notation.Note: If the answer includes more than one interval write theintervals separated by the union symbol, U. If needed enter −∞


• (-infinity,infinity)

12. (5 pts) Library/maCalcDB/setAlgebra15Functions/ns1 2 11.pgAt the surface of the ocean, the water pressure is the same asthe air pressure above the water, about 15 lb/in2, Below thesurface the water pressure increases by about 5.34 lb/in2 forevery 10 ft of descent.Write a function f (x) which expresses the water pressure inpounds per square inch as a function of the depth in inches be-low the ocean surface.f (x) =At what depth is the pressure 110 lb/in2? Include the units inyour answer:

Correct Answers:• 0.0445*x +15• 2134.83146067416 in

13. (5 pts) maCalcDB/setAlgebra15Functions/beth7fixed.pgList all real values of x such that f (x) = 0. If there are no suchreal x, type none in the answer blank. If there is more that onereal x, give a comma separated list, e.g. 1,2

f (x) =−18x2 +3x+19

x =Correct Answers:

• -0.947443073071082, 1.11410973973775

14. (5 pts) Library/Rochester/setAlgebra15Functions/srw2 1 25.pgGiven the function f (x) =−7+ x2, calculate the following val-ues:f (x+1) =f (x)+ f (−4) =

Correct Answers:• -7+(x+1)**2

2

• -7+x**2+-7+(-4)**2

15. (5 pts) Library/ma112DB/set6/sw4 2 1.pgClick on the graph to view a larger graphFor the function h(x) given in the graph

its domain is [ , ];its range is [ , ];and then enter the corresponding function value in each answerspace below:

1. h(−2)2. h(0)3. h(3)4. h(−3)

Correct Answers:

• -3• 3• -2• 2• 1• -2• -1• 2

16. (5 pts) Library/ma112DB/set6/c4s2p5 7/c4s2p5 7.pgClick on the graphs to view enlarged onesEnter Yes or No in each answer space below to indicate whetherthe corresponding curve defines y as a function of x.

1.

2.

3.

4.

3

5.

6.Correct Answers:

• Yes• Yes• Yes• No• No• Yes

17. (5 pts) Library/ma112DB/set6/c4s2p19 40/c4s2p19 40.pgMatch the functons with their graphs. Enter the letter of thegraph below which corresponds to the function. (Click on im-age for a larger view )

1. 32. 1

x3. |x|+ x+14. |2x|

A B C D

Correct Answers:

• C• A• B• D

18. (5 pts) Library/ma112DB/set6/c4s2p59 72/c4s2p59 72.pgMatch the functons with their graphs. Enter the letter of thegraph below which corresponds to the function. (Click on im-age for a larger view )

1. Piecewisefucntion : f (x) = x, ifx ≤ 0and f (x) = x +1, ifx > 0

2. Piecewisefucntion : f (x) = −1, ifx < 2and f (x) =1, ifx ≥ 2

3. Piecewisefucntion : f (x) = 1, ifx ≤ 1and f (x) = x +1, ifx > 1

4. Piecewisefucntion : f (x) = 3, ifx < 2and f (x) = x −1, ifx ≥ 2

A B C D

Correct Answers:

• D• B• A• C

19. (5 pts) Library/Rochester/setAlgebra16FunctionGraphs/c0s1p1-/c0s1p1.pgThe simplest functions are the linear (or affine) functions — thefunctions whose graphs are a straight line. They are importantbecause many functions (the so-called differentiable functions)“locally” look like straight lines. (“locally” means that if wezoom in and look at the function at very powerful magnificationit will look like a straight line.)

Enter the letter of the graph of the function which corre-sponds to each statement.

1. The graph of the line is increasing2. The graph of the line is decreasing3. The graph of the line is constant4. The graph of the line is not the graph of a function

A B C D

Correct Answers:

• C• D• B• A

4

20. (5 pts) Library/Rochester/setAlgebra16FunctionGraphs/c0s1p3-/c0s1p3.pg20. (5 pts) Library/Rochester/setAlgebra16FunctionGraphs/c0s1p3-/c0s1p3.pgAlmost any kind of quantitative data can be represented by agraph and most of these graphs represent functions. This is whyfunctions and graphs are the objects analyzed by calculus. Thenext two problems illustrate data which can be represented bya graph. Match the following descriptions with their graphs be-low:

1. The graph of the distance traveled by a car as it drivesalong a city street vs. time.

2. The graph of the distance traveled by a car as it enters asuperhighway vs. time.

3. The graph of the velocity of a car as it drives along acity street vs. time.

4. The graph of the velocity of a car entering a superhigh-way vs. time.

A B C D

Correct Answers:

• B• A• C• D

21. (5 pts) Library/ASU-topics/setFunctions/rich2.pgTemperature readings T (in degrees Fahrenheit) were recordedevery 2 hours from midnight to noon in Tempe, Arizona on acool December morning. The time t was measured in hoursafter midnight.

time, t Temp, T0 492 474 446 458 4710 5012 51

Which graph best represents the data?Note: Click on any graph to view a larger graph.

• A.

• B.

• C.

5

• D.

• E.

• F.

Correct Answers:• C

22. (1 pt) 1060Library/set3 Algebra/s3p16.pg

A function f is even if it satisfies f (x) = f (−x) for all x inits domain. An example of an even function is f (x) = x2 since(x2) = (−x)2.

f is odd if it satisfies f (x) =− f (−x) for all x in its domain. Anexample of an odd function is f (x) = x3 since x3 =−(−x)3.

Functions may be neither even nor odd, for example the functionf (x) = x2 + x3 is in that category.

For each function below enter the letter E if the function iseven, the letter O (not the digit 0!) if it’s odd, and the letter N ifit’s neither even nor odd.

f (x) = x4.f (x) = x5.f (x) = x4 + x5.

Correct Answers:

• E• O• N

23. (1 pt) 1060Library/set3 Algebra/s3p17.pg

For each function below enter the letter E if the function iseven, the letter O if it’s odd, and the letter N if it’s neither evennor odd.

f (x) = x4 + x2.f (x) = x5 + x3.f (x) = x4 + x5.f (x) = |x|.f (x) = |x|3.f (x) = x2

x2+1 .

HINT: If in doubt, evaluate the functions at pairs of numberslike 1 and and-1).

Correct Answers:

• E• O• N• E• E• E

24. (1 pt) 1060Library/set3 Algebra/s3p18.pgSuppose f is a function that’s defined for all real numbers x.Assume it’s neither even nor odd. Below you will find otherfunctions defined in terms of f

For each function enter the letter E if the function is even, theletter O if it’s odd, and the letter N if it’s neither even nor odd.

g(x) = f (x)+ f (−x).g(x) = f (x)− f (−x).g(x) = f (−x).

Correct Answers:

• E• O• N

6


7

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Prep Lines is due : 09/08/2008 at 12:00pm MDT.The

(* replace with url for the course home page *)for the course contains the syllabus, grading policy and other information.

This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.




1. (9 pts) Library/Rochester/setAlgebra14Lines/srw1 10 19a-sol.pgThe equation of the line that goes through the point (5,5) and isperpendicular to the line 2x +4y = 3 can be written in the formy = mx+b where m is:and where b is:

Correct Answers:

• 2• -5

2. (9 pts) Library/ASU-topics/setLinesInPlane/bethlnint1.pgA line through (-9, -8) with a slope of -1 has a y-intercept at

Correct Answers:

• (0,-17)

3. (9 pts) Library/ASU-topics/setLinesInPlane/sApB 31-36.pgAn equation of a line through (3, 3) which is parallel to the liney = 4x+2 has slope:

and y-intercept at:

Correct Answers:

• 4• -9

4. (9 pts) Library/ASU-topics/setLinesInPlane/lines2.pg

What is the slope of the line through (7, 2) and (7,-3)? If theslope is undefined, type undefined .

What is the slope of the line through (-6, -6) and (-5,-7)? Ifthe slope is undefined, type undefined .

What is the slope of the line through (5, 2) and (4,2)? If theslope is undefined, type undefined .

Correct Answers:

• undefined• -1• 0

5. (9 pts) Library/ASU-topics/setLinesInPlane/sApB 31-36a.pgFind the equation of a line through (-4, 6) which is perpendicu-lar to the line y = 3x+1, and put it in the form y = mx+b.m =b =

Correct Answers:

• -0.333333333333333• 4.66666666666667

6. (9 pts) Library/ASU-topics/setLinesInPlane/bethpt1.pgAn equation of a line through (-3, 3) which is perpendicular tothe line y = 2x+1 has slope:

and y-intercept at:

Correct Answers:

• -0.5• 1.5

7. (9 pts) Library/ASU-topics/setLinesInPlane/srw1 10 41.pgFind the slope and y-intercept of the line 18x+19y = 0.The slope of the line is:The y-intercept of the line is:

Correct Answers:

• -0.947368421052632• 0

1

8. (9 pts) Library/ASU-topics/setLinesInPlane/sApB 7-10.pgFind the slope of the line through (7, 0) and (-4, 3).

Correct Answers:

• -0.272727272727273

9. (9 pts) Library/Rochester/setAlgebra07PointsCircles/srw1 9 4-sol.pgConsider the two points (2,−3) and (−6,−4). The distancebetween them is:The x co-ordinate of the midpoint of the line segment that joinsthem is:The y co-ordinate of the midpoint of the line segment that joinsthem is:

Correct Answers:

• 8.06225774829855• -2• -3.5

10. (9 pts) Library/Rochester/setAlgebra07PointsCircles/srw1 9 2-sol.pgConsider the two points (1,−2) and (7,8). The distance be-tween them is:The x co-ordinate of the midpoint of the line segment that joinsthem is:The y co-ordinate of the midpoint of the line segment that joinsthem is:

Correct Answers:

• 11.6619037896906• 4

• 3

11. (18 pts) Library/Rochester/setAlgebra07PointsCircles-/equidist off axis.pgFind the point (x,y) on the line y = x that is equidistant from thepoints (−4,9) and (−5,−7).

x =y =

Correct Answers:• 0.676470588235294• 0.676470588235294

12. (9 pts) Library/Rochester/setAlgebra07PointsCircles/sApB x-sol.pgFind the perimeter of the triangle with the vertices at(3,0), (−5,6), and (−2,−3).

Correct Answers:• 25.3177848753504

13. (9 pts) Library/Rochester/setAlgebra07PointsCircles/srw1 9 4.pgConsider the two points (4,−4) and (−6,−6). The distancebetween them is:The x co-ordinate of the midpoint of the line segment that joinsthem is:The y co-ordinate of the midpoint of the line segment that joinsthem is:

Correct Answers:• 10.1980390271856• -1• -5


2

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Prep More Functions is due : 09/15/2008 at 12:00pm MDT.The


This assignment deals with functions. The relevant material can be found in sections 1.7-1.9 of the text.




1. (12 pts) Library/SUNYSB/WebWorkInverse2.pgIf f is one-to-one and f (6) = 12, thenf−1(12) =

and ( f (6))−1 = .

If g is one-to-one and g(−7) = 13, theng−1(13) =

and (g(−7))−1 = .

If f is one-to-one and f (−15) = 6, then f−1(6) = and( f (−15))−1 = .

If g is one-to-one and g(−6) = 12, then g−1(12) = and(g(−6))−1 = .

If f (x) = 6x−2, thenf−1(y) =f−1(−6) =

Correct Answers:

• 6• 0.0833333333333333• -7• 0.0769230769230769• -15• 0.166666666666667• -6• 0.0833333333333333• (y+2)/6• -0.666666666666667

2. (12 pts) SUNYSB/WebWorkInverse.pgLet

f : R−→ R, f (x) = 2− x

f−1(x) =Let

f : [0,∞)−→ R, f (x) = 3− x2

f−1(x) =Let

f : C −→ D, f (x) =x+4x+7

f is bijective from C to D.f−1(−10) =Let

f (x) =15

x−1, −3≤ x ≤ 9

The domain of f−1 is the interval [A,B]where A = and where B =If f : R−→ R, f (x) = 4x−13, then

f−1(y) =f−1(−6) =

Correct Answers:

• 2 - x• (3 - x)ˆ(1/2)• -6.72727272727273• -1.6• 0.8• (y+13)/4• 1.75

3. (12 pts) Library/SUNYSB/inverseFunction.pgFor each of the following, find the inverse f−1(x) of the func-tion. You can assume that f : R −→ R. If there is no inverse,enter -1.

Hint: you can write√

x as x carat (1/2) where carat is shift-6on most keyboards.

The inverse of f (x) = x+54 is

The inverse of f (x) = 5x+2 is1

The inverse of f (x) = 6x isThe inverse of f (x) = x2 isThe inverse of f (x) = x

13 is

The inverse of f (x) = x5 isCorrect Answers:

• x*(4)-(5)• (x-(2))/(5)• x/(6)• -1• xˆ3• xˆ(0.2)

4. (12 pts) Library/ASU-topics/setInverseFunctions/garcia1.pg

Consider the graphs below.

Determine if the function in the graph is one-to-one.

• A. The function is one-to-one.• B. The function is not one-to-one.





2





Correct Answers:

• B

• A• A• B• B

5. (12 pts) Library/ASU-topics/setInverseFunctions/bethinvfun1.pgAlgebraically find the inverse function of f (x) = 9x+8.f−1(x) =

Correct Answers:• (x-8)/9

6. (12 pts) Library/SUNYSB/functionComposition.pgf and g are functions from R to R.Consider f (x) = 4x+3,g(x) = x7.f ◦g =

g◦ f =

Consider f (x) =√

x2 +7,g(x) = x2 +7.f ◦g =

g◦ f =Correct Answers:

• (4)*xˆ(7)+3• ((4)*x+3)ˆ(7)• ((xˆ2+(7))ˆ2 + (7))ˆ0.5• xˆ2+(7)+(7)

7. (12 pts) Library/ma112DB/set7/sw4 7 1.pgFor this question, input infinity for ∞ and input -infinity for−∞.Given that f (x) = x2−14x and g(x) = x+6, find(a) f +g= and its domain is ( , )(b) f −g= and its domain is ( , )(c) f g= and its domain is ( , )(d) f /g= and its domain is x 6=

Correct Answers:• x**2-(14-1)*x+6• -infinity• infinity• x**2-(14+1)*x-6• -infinity• infinity• (x**2-14*x)*(x+6)• -infinity• infinity• (x**2-14*x)/(x+6)• -6

8. (12 pts) Library/Rochester/setAlgebra17FunComposition-/srw2 8 17.pgGiven that f (x) = 9x−5 and g(x) = 8− x2, calculate(a) f (g(0))=(b) g( f (0))=

Correct Answers:• 67

3

• -17

9. (12 pts) Library/Rochester/setAlgebra17FunComposition-/ns1 2 29.pgLet f (x) = x3 +4x2 and g(x) = 7x2−4.

f /g is undefined at two points A and B where A < B.A is ,and B is

Correct Answers:• -0.755928946018454• 0.755928946018454

10. (12 pts) Library/SUNYSB/WebWorkInverse.pgLet

f : R−→ R, f (x) = 9− xf−1(x) =Let

f : [0,∞)−→ R, f (x) = 3− x2

f−1(x) =Let

f : C −→ D, f (x) =x+5x+6

f is bijective from C to D.f−1(−5) =Let

f (x) =16

x−3, −5≤ x ≤ 5

The domain of f−1 is the interval [A,B]where A = and where B =Let

f : R−→ R, f (x) = 3+3x+4ex

f−1(7) =If f : R−→ R, f (x) = 7x−5, then

f−1(y) =f−1(8) =

Correct Answers:• 9 - x• (3 - x)ˆ(1/2)• -5.83333333333333• -3.83333333333333• -2.16666666666667• 0• (y+5)/7• 1.85714285714286


4

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Prep Angles is due : 09/22/2008 at 12:00pm MDT.The


This assignment covers sections 4.1 and 4.2 of the text.




1. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p1.pgWe need to be able to convert radians to degrees and vice versa.Below you can enter radians as decimal approximations, but Irecommend that you enter arithmetic expressions like pi/2 forπ

2 . I trust that you can compute a decimal value on your calcu-lator if required. However, if you do enter a decimal approxi-mation compute and enter at least four digits. WeBWorK willconsider your value correct if it is within one tenth of one per-cent of the answer that it has been told.

In this problem WeBWorK will tell you for each answerwhether it’s right or wrong, in the next problem you will have toget everything right before getting credit.

90◦ = rad60◦ = rad85◦ = rad−579◦ = rad

Hint: Everything follows from the fact that

360◦ = 2π rad.

(Show hint after 1 attempts. )

Correct Answers:

• 1.5707963267949• 1.0471975511966• 1.48352986419518• -10.1054563690472

2. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p2.pgFill in the following equations:

45◦ = rad266◦ = rad−658◦ = rad


360◦ = 2π rad.


Correct Answers:

• 0.785398163397448• 4.64257581030492• -11.4842664781227

3. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p3.pgIn this problem you are asked to convert radians to degrees.

πrad = ◦.1.5rad = ◦.−4πrad = ◦.Hint: Everything follows from the fact that

360◦ = 2πrad.

Correct Answers:

• 180• 85.9436692696235• -720

4. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p4.pgIn this problem you are asked to convert radians to degrees.

6πrad = ◦.−0.7rad = ◦.(√

2)

πrad = ◦.


360◦ = 2π rad.

(Show hint after 1 attempts. )1

Correct Answers:

• 1080• -40.1070456591576• 254.558441227157

5. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p5.pgOccasionally you may encounter an angle that’s given in the oldfashioned degrees/minutes/second form, as described in the boxon page 266 of the textbook (6th edition, for other editions itmay be a different page). A degree is divided into 60 minutes(or minutes of degree), and a minute (of degree) is divided into60 seconds.

Leta = 3◦5′17′′.

Then a = degrees (in decimal notation)and a = radians.Again, enter your answers showing at least 4 digits.

Hint: One minute is 1/60 of a degree, and 1 second is 1/3600of a degree.


Correct Answers:

• 3.08805555555556• 0.0538967369289471

6. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p6.pgIt’s a little trickier to convert an angle that’s given in decimalnotation into the minute/second notation. Let’s do it just once.The angle a = 6.75444... degrees can be written asa = ◦ ′ ′′.

Hint: We need to convert .75444... into minutes and sec-onds. Recall that one minute equals 1/60 degrees. See howmany minutes fit into .75444... and then see how many secondsfit in what’s left.


Correct Answers:

• 6• 45• 16

7. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p9.pg

You travel again in your trusty car with the 25 inch wheels.You installed a nifty gadget that measures how often yourwheels turn each hour. It shows that right now they turn 49000times per hour. You figure you are going at a speed of

miles per hour. (Round your answer to the nearest tenth of amile per hour.)

Hint: Multiply the circumference of the wheel with the numberof rotations per hour. Convert inches to miles.


Correct Answers:

• 60.7


This problem outlines how Eratosthenes of Cyrene approx-imated the diameter of the earth in approximately 200BC. Byobserving the shadow of the sun at noon he recognized that hishome town of Alexandria was approximately 7.2◦ north of thetown of Syene. He paid somebody to walk and measure thedistance between Syene and Alexandria. Suppose it is 787 km.Based on these figures, the circumference of the earth iskilometers.

Hint: Observe that 7.2◦ is to 360 degrees as walking 787km is to walking the entire circumference.


Correct Answers:

• 39350

9. (10 pts) set math1060fall2005-1/set4/s4p11.pgRecall than an angle is acute if it’s between 0 and π

2 radians,right if it equals π

2 radians, and obtuse if is between π

2 and π

radians. For each angle below enter R if it is a right angle, A ifit’s an acute angle, and O (upper case o, not the digit 0) if it isan obtuse angle.

90◦

60◦

130◦

1rad2radπ

2 radCorrect Answers:

• R• A• O• A• O• R

2

10. (10 pts) 1060Library/set math1060fall2005-1/set4/s4p13.pgThe remaining problems in this set deal with the definitions ofthe basic trigonometric functions, sin, cos, and tan.

You can answer some of these questions simply by keying thingsinto your calculator. However, the purpose of these problems isto help you get familiar with the definitions of the basic trigono-metric functions, and to improve your ability to work with thosedefinitions. All the questions can be answered straight from thedefinitions of the trigonometric functions, perhaps after drawinga simple picture, without the aid of a calculator, and I recom-mend that you don’t use one. Use ’pi’ to enter the value of π

and use sqrt(...) to enter the square root of something.

A line drawn from the origin and forming the angle of t = 7π

6with the positive x-axis intersects the unit circle at the point(−√

32 ,− 1

2

). Complete the following equations:

t = degrees.

cos t = .

sin t = .

tan t = .Correct Answers:

• 210• -0.866025403784439• -0.5• 0.577350269189626


A line drawn from the origin and forming the angle t with thex-axis intersects the unit circle at the point

(13 , 2

√2

3

). Complete

the following equations:

cos t = .sin t = .tan t = .

Correct Answers:• 0.333333333333333• 0.942809041582063• 2.82842712474619



(√5

3 ,− 23

). Complete



Correct Answers:• 0.74535599249993• -0.666666666666667• -0.894427190999916


Let t be the angle between 0 and π

2 such that

sin t =14.

Then

cos t = .sin(−t) = .cos(−t) = .tan t = .

Correct Answers:• 0.968245836551854• -0.25• 0.968245836551854• 0.258198889747161


3

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Transformations and Combinations of Functions is due : 09/24/2008 at 03:00pm MDT.The


This assignment deals with transformations and combinations of functions. This material can be found in sections 1.7-1.9 of thetext.




1. (11 pts) 1050Library/set4 Functions and Their Graphs-/1050s4p13.pg

The circle in the graph above is described by the equation

(x−h)2 +(y− k)2 = r2

whereh = ,k = , andr = .

The graph is not very detailed, but all the answers in thisquestion are integers. The screen version of the graph is largerthan the printed version. In this problem, WeBWorK will tellyou for each partial answer whether it us correct or not.

Correct Answers:

• 2

• 1• 3


The circle in the graph above is described by the equation

(x−h)2 +(y− k)2 = r2

whereh = ,k = , andr = .

Correct Answers:

• -2• 3• 2

1


A.

B.

C.

Match the graphs above with the equations below. (Enter A, B,and C, as appropriate.)Clicking on the above graphs will show you a larger version ofthe same graph.

y = x+1 ,x2 + y2 = 1 , andy = x2 .

Correct Answers:

• C• A• B

4. (11 pts) 1050Library/set5 Functions and Their Graphs-/1050s5p18.pgSuppose

f (x) = x2 + x.

Then

f (1) = ,

f (x2) = , and

f ( f (x)) = .

Correct Answers:

• 2• xˆ4+xˆ2• x**4+2*x**3+2*x**2+x

2

5. (11 pts) 1050Library/set5 Functions and Their Graphs-/1050s5p19.pgThe next few problems are focused on the combination of func-tions: Suppose f and g are two functions. Then we define

A. ( f +g)(x) = f (x)+g(x),B. ( f −g)(x) = f (x)−g(x),C. ( f g)(x) = ( f ×g)(x) = f (x)×g(x),D. ( f ÷g)(x) = ( f /g)(x) =

(fg

)(x) = f (x)

g(x) ,

E. ( f ◦g)(x) = f (g(x)), andF. (g◦ f )(x) = g( f (x)).

Supposef (x) = 2x+1

andg(x) = x+2.

Then

1. ( f +g)(x) = ,2. ( f −g)(x) = ,3. ( f g)(x) = ,4. ( f /g)(x) = ,5. ( f ◦g)(x) = , and6. (g◦ f )(x) = .

Correct Answers:• 3x+3• x-1• 2*x**2+5*x+2• (2x+1)/(x+2)• 2x+5• 2x+3


f (x) = x2 +1and

g(x) = x+1.

Then

A. ( f +g)(x) = ,B. ( f −g)(x) = ,C. ( f g)(x) = ,D. ( f /g)(x) = ,E. ( f ◦g)(x) = , andF. (g◦ f )(x) = .

Correct Answers:• xˆ2+x+2

• xˆ2-x• xˆ3+xˆ2+x+1• (xˆ2+1)/(x+1)• xˆ2+2x+2• xˆ2+2


f (x) =x

x+1

and

g(x) =x+1

x.

Then

(( f ◦g)− (g◦ f ))(x) = .Correct Answers:

• -(3xˆ2+3x+1)/(x(2x+1))

8. (11 pts) 1050Library/set math1050fall2002-2/set3/pr9.pg

Match each graph to its equation.(For all graphs on this page, if you are having a hard time seeingthe picture clearly, click on it. It will expand to a larger pictureon its own page so that you can inspect it more closely.)

1.3

2.

3.

4.

5.

6.

A. x2 = 2yB. (x−1)2 =−2(y−1)C. (y−1)2 =−2(x−1)D. x2 =−2yE. (y−1)2 = 2(x+1)F. y2 = 2x

Correct Answers:

• A• F• E• B• D• C

9. (11 pts) 1050Library/set math1050fall2002-2/set3/pr27.pgLet g be the function below.

For all graphs on this page, if you are having a hard timeseeing the picture clearly, click on it. It will expand to a largerpicture on its own page so that you can inspect it more closely.

4

The domain of g(x) is of the form [a,b], where a is and bis .

The range of g(x) is of the form [c,d], where c is and dis .

Enter the letter of the graph which corresponds to each newfunction defined below:

1. g(x−2)+2 is .2. g(2x) is .3. 2+g(−x) is .4. g(x+2)−2 is .

A B C D

2.71828182845905 F G H

Correct Answers:

• -3• 3• 0• 2• B• G• E• D

10. (11 pts) 1050Library/set math1050fall2002-2/set3/pr33.pgBelow is the graph of a function f : (Click on image for a largerview )

Graph A

Graph B

5

Graph CThe inverse of the function f is (A, B or C):Correct Answers:

• A

11. (11 pts) 1050Library/set math1050fall2002-2/set3/pr34.pgBelow is the graph of a function f : (Click on image for a largerview )

Graph A

Graph B

Graph C

6

Graph D

The inverse of the function f is (A, B, C or D):Correct Answers:

• B


7

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Prep Trig Functions is due : 09/29/2008 at 12:00pm MDT.The


This Prep assignment is about trig function. You can find this material in sections 4.1-4.4 of the text




1. (12 pts) 1060Library/set5 Trigonometry/s5p1.pgYou may think this first problem a little odd. All it asks is thatuse your calculator to evaluate the trigonometric functions ofsome angles. The reason for including this problem is the ob-servation that a substantial number of mistakes in solving trigproblems are due to having your calculator in the wrong mode,and not noticing this fact. Whenever you use your calculatorcheck its mode!

In this problem you will need to enter your answers as decimalapproximations (with at least 4 digits). Usually WeBWorK willlet you enter expressions like sin(...), but note that in that caseWeBWorK always assumes your angles are measured in radians.

sin(15◦) =sin(15) =cos(1.4◦) =cos(1.4) =

Correct Answers:

• 0.258819045102521• 0.650287840157117• 0.999701489781183• 0.169967142900241

2. (12 pts) 1060Library/set5 Trigonometry/s5p2.pgThe next few problems deal with the definition of the trigono-metric functions in a right triangle. We use a standard notation:a, b, and c denote the sides of the triangles and their lengths, andA, B, and C denote the angles opposite a, b, and c, respectively,as indicated in the nearby figure.

Thus a and b are the lengths of the two short sides, and c isthe hypotenuse. Use the Pythagorean Theorem to compute anymissing length. Assume also that A denotes the angle oppositea, B the angle opposite b, and, of course, C the right angle.

I recommend that in this problem you enter values of the trigfunctions as fractions.

Suppose a = 5 and b = 12.Then

c = ,sin(A) = ,cos(A) = , andtan(A) = .

Correct Answers:

• 13• 0.384615384615385• 0.923076923076923• 0.416666666666667

1

3. (12 pts) 1060Library/set5 Trigonometry/s5p6.pgYou are flying a kite on a line that is 350 feet long. Let’s sup-pose the line is perfectly straight (it never really is) and it makesan angle of 65 degrees with the horizontal direction. The kite isflying at an altitude of feet.Hint: Draw a picture. Look for a right triangle.

Correct Answers:

• 317.2077

4. (12 pts) set5 Trigonometry/s5p11.pgRecall the definition: Let θ be an angle in standard position.Its reference angle is the acute angle θ′ formed by the terminalside of θ and the horizontal axis.

For example, reference angle of 150 degrees is 30 degrees (180-150=30), the reference angle of 205 degrees is 25 degrees (205-180=25), the reference angle of 13pi/6 is pi/6 (13pi/6-2pi=pi/6).

Below, express θ′ in the same units (degrees or radians) as θ.You can enter arithmetic expressions like 210-180 or 3.5-pi.

The reference angle of 100◦ is ◦.The reference angle of 350◦ is ◦.The reference angle of 4 is .Hint: Draw the angle. The Figures on page 314 of the textbookmay be helpful

Correct Answers:

• 80• 10• 0.858407346410207

5. (12 pts) 1060Library/set5 Trigonometry/s5p12.pg

Below, express the reference angle θ′ in the same units (degreesor radians) as θ. You can enter arithmetic expressions like 210-180 or 3.5-pi.

The reference angle of 30◦ is ◦.The reference angle of −30◦ is ◦.The reference angle of 1,000,000◦ is ◦.The reference angle of 100 is .Hint: Draw the angle. The Figures on page 314 of the textbookmay be helpful. To see the angle 1,000,000◦ subtract a suit-able multiple of 360◦. To see the angle 100, subtract a suitablemultiple of 2π.

Correct Answers:

• 30• 30• 80• 0.53096491487338

6. (12 pts) 1060Library/set5 Trigonometry/s5p13.pg

You approach a hill on top of which there is a tall radio an-tenna. You know from your map that your horizontal distancefrom the bottom of the radio antenna is 600 feet. The angle ofelevation to the bottom of the antenna is 10◦, and the angle ofelevation to the top of the antenna is 25◦. You figure that theheight of the hill is feet, and the height of the antenna is

feet. (Enter your answers rounded to the nearest foot.)Hint: Draw a picture. Figure out the height of the hill. Figureout the combined height of the antenna and the hill. Computethe difference.

Correct Answers:

• 105.796188425079• 173.98840646792

7. (12 pts) 1060Library/set6 Trigonometry/q1.pg

Letf (x) = 3sin(5x−4)+2.

The amplitude of this function is ,its period is ,its phase shift is ,and its vertical translation is ,

Correct Answers:

• 3• 1.25663706143592• 0.8• 2


Letf (x) = 7sin(3πx−2)−3.


Correct Answers:

• 7• 0.666666666666667• 0.212206590789194• -3

9. (12 pts) 1060Library/set6 Trigonometry/q3.pgLet

f (x) = 12sin(4x−2)+3.


Correct Answers:

• 12• 1.5707963267949• 0.5

2

• 3



f (x) = sinx.f (x) = cosx.f (x) = tanx.f (x) = (sinx)(cosx).f (x) = sin2 x.

Note: Remember that this notation means the square of sinx,

i.e.,

sin2 x = (sinx)2.

f (x) = cos2 xf (x) = sinx+ cosx

Correct Answers:

• O• E• O• O• E• E• N


3

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Angles and Trig Functions is due : 10/01/2008 at 03:00pm MDT.The






1. (15 pts) 1060Library/set5 Trigonometry/s5p2.pgThe next few problems deal with the definition of the trigono-metric functions in a right triangle. We use a standard notation:a, b, and c denote the sides of the triangles and their lengths, andA, B, and C denote the angles opposite a, b, and c, respectively,as indicated in the nearby figure.




c = ,sin(A) = ,

cos(A) = , andtan(A) = .

Correct Answers:

• 13• 0.384615384615385• 0.923076923076923• 0.416666666666667

2. (15 pts) 1060Library/set5 Trigonometry/s5p3.pgUsing the same notation as in the preceding problem, supposeb = 3 and c = 5.

Thena = ,sin(A) = ,cos(A) = , andtan(A) = .

Correct Answers:

• 4• 0.8• 0.6• 1.33333333333333

3. (15 pts) 1060Library/set5 Trigonometry/s5p4.pgUsing the same notation as in the preceding problem, supposea = 20 and c = 29.

Thenb = ,sin(B) = ,cos(A) = , andtan(A) = .

Correct Answers:

• 21• 0.724137931034483• 0.724137931034483• 0.952380952380952

1

4. (15 pts) 1060Library/set5 Trigonometry/s5p5.pgYou are hiking along the edge of the Green River (which is run-ning straight in the area of interest). Straight across from youon the opposite shore there is a particularly noticeable boulderat the edge of the river. You walk 120 feet along the shore ofthe river, and now the line from you to the boulder makes anangle of 35◦ with the edge of the river. You whip out your trustyscientific calculator and deduce that the width of the river at thispoint is feet.

Hint: Draw a picture. ”Straight across” means that the line fromyou to the boulder forms a right angle with the shore of the river.


5. (15 pts) 1060Library/set5 Trigonometry/s5p7.pgSuppose you have an isosceles triangle, and each of the equalsides has a length of 1 foot. Suppose the angle formed by thosetwo sides is 45◦. Then the area of the triangle issquare feet.

Hint: Draw the triangle. Draw the height. It divides the tri-angle into two congruent right triangles. Compute the missinglengths of the sides of that triangle using the trigonometric func-tions. Then compute the area of one of those two triangles, andmultiply with 2.


6. (15 pts) 1060Library/set5 Trigonometry/s5p9.pgThis is a generalization of the previous problem. The ideas areexactly the same.

Suppose you have an isosceles triangle, and each of the equalsides has a length of r feet. Suppose the angle formed by thosetwo sides is t. Then the area of the triangle issquare feet.

Hint: Draw the triangle. Draw the height. It divides the tri-angle into two congruent right triangles. Compute the missing

lengths of the sides of that triangle using the trigonometric func-tions. Then compute the area of one of those two triangles, andmultiply with 2.

Correct Answers:

• r**2*sin(t/2)*cos(t/2)

7. (15 pts) 1060Library/set5 Trigonometry/s5p10.pgSuppose you inscribe a regular octagon into a circle of radius r.Then the area of that octagon is .

Hint: Use the result from the preceding problem.Correct Answers:

• 8*r**2*sin(0.785398163397448/2)*cos(0.785398163397448/2)

8. (15 pts) 1060Library/set4 Trigonometry/s4p7.pgThe diameter of the wheels on your car (including the tires) is25 inches. You are going to drive 239 miles today. Each of yourwheels is going to turn by an angle of degrees.

Hint: A mile has 5280 feet, a foot has 12 inches, and the cir-cumference of a circle with a diameter d equals πd.

Correct Answers:

• 69410582

9. (1 pt) setPrep Dot Product/prob5.pgCompute

|| 〈10,1〉 ||

Correct Answers:

• 10.0498756211209

10. (1 pt) setPrep Dot Product/prob8.pg

u = 〈−2,−5〉Compute

||u||

Correct Answers:

• 5.3851648071345


2

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Chapter 4 exam practice is due : 10/03/2008 at 08:00am MDT.The






1. (1 pt) 1060Library/set4 Trigonometry/s4p2.pgFill in the following equations:

45◦ = rad168◦ = rad−600◦ = radHint: Everything follows from the fact that

360◦ = 2π rad.

Correct Answers:

• 0.785398163397448• 2.93215314335047• -10.471975511966

2. (1 pt) 1060Library/set4 Trigonometry/s4p3.pgIn this problem you are asked to convert radians to degrees.

πrad = ◦.1.5rad = ◦.−4πrad = ◦.Hint: Everything follows from the fact that

360◦ = 2π rad.

Correct Answers:

• 180• 85.9436692696235• -720

3. (1 pt) 1060Library/set4 Trigonometry/s4p7.pgThe diameter of the wheels on your car (including the tires) is25 inches. You are going to drive 474 miles today. Each of yourwheels is going to turn by an angle of degrees.

Hint: A mile has 5280 feet, a foot has 12 inches, and the cir-cumference of a circle with a diameter d equals πd.

Correct Answers:

• 137659482

4. (1 pt) 1060Library/set4 Trigonometry/s4p11.pgRecall than an angle is acute if it’s between 0 and π

2 radians,right if it equals π

2 radians, and obtuse if is between π

2 and π

radians. For each angle below enter R if it is a right angle, A ifit’s an acute angle, and O (upper case o, not the digit 0) if it isan obtuse angle.

90◦

60◦

130◦

1rad2radπ

2 radCorrect Answers:

• R• A• O• A• O• R

5. (1 pt) 1060Library/set4 Trigonometry/s4p13.pgThe remaining problems in this set deal with the definitions ofthe basic trigonometric functions, sin, cos, and tan.

You can answer some of these questions simply by keying thingsinto your calculator. However, the purpose of these problems isto help you get familiar with the definitions of the basic trigono-metric functions, and to improve your ability to work with thosedefinitions. All the questions can be answered straight from thedefinitions of the trigonometric functions, perhaps after drawinga simple picture, without the aid of a calculator, and I recom-mend that you don’t use one. Use ’pi’ to enter the value of π

and use sqrt(...) to enter the square root of something.

A line drawn from the origin and forming the angle of t = 7π

61

with the x-axis intersects the unit circle at the point(−√

32 ,− 1

2

).

Complete the following equations:t = degrees.

cos t = .

sin t = .

tan t = .Correct Answers:

• 210• -0.866025403784439• -0.5• 0.577350269189626

6. (1 pt) 1060Library/set4 Trigonometry/s4p14.pg


(13 , 2

√2

3

). Complete



Correct Answers:

• 0.333333333333333• 0.942809041582063• 2.82842712474619

7. (1 pt) 1060Library/set5 Trigonometry/s5e2.pgThe next few problems deal with the definition of the trigono-metric functions in a right triangle. In all cases assume a and bare the lengths of the two short sides, and c is the hypotenuse.Use the Pythagorean Theorem to compute the missing lengths.Assume also that A denotes the angle opposite a, B the angleopposite b, and, of course, C the right angle. We are not in thatchapter, yet, but Figure 4.67 on page 422 illustrates that conven-tion.

sin(15◦) =sin(15) = radcos(1.4◦) =cos(1.4) = rad

Correct Answers:

• 0.258819045102521• 0.650287840157117• 0.999701489781183• 0.169967142900241

8. (1 pt) 1060Library/set5 Trigonometry/s5p2.pgThe next few problems deal with the definition of the trigono-metric functions in a right triangle. We use a standard notation:a, b, and c denote the sides of the triangles and their lengths, and

A, B, and C denote the angles opposite a, b, and c, respectively,as indicated in the nearby figure.




c = ,sin(A) = ,cos(A) = , andtan(A) = .

Correct Answers:• 13• 0.384615384615385• 0.923076923076923• 0.416666666666667

9. (1 pt) 1060Library/set5 Trigonometry/s5p6.pgYou are flying a kite on a line that is 350 feet long. Let’s sup-pose the line is perfectly straight (it never really is) and it makesan angle of 65 degrees with the horizontal direction. The kite isflying at an altitude of feet.Hint: Draw a picture. Look for a right triangle.


10. (1 pt) 1060Library/set5 Trigonometry/s5p13.pg

You approach a hill on top of which there is a tall radio an-tenna. You know from your map that your horizontal distancefrom the bottom of the radio antenna is 600 feet. The angle ofelevation to the bottom of the antenna is 10◦, and the angle ofelevation to the top of the antenna is 25◦. You figure that theheight of the hill is feet, and the height of the antenna is

2

feet. (Enter your answers rounded to the nearest foot.)Hint: Draw a picture. Figure out the height of the hill. Figureout the combined height of the antenna and the hill. Computethe difference.

Correct Answers:• 105.796188425079• 173.98840646792

11. (1 pt) 1060Library/set6 Trigonometry/q1.pg

Letf (x) = 3sin(5x−4)+2.


Correct Answers:• 3• 1.25663706143592• 0.8• 2



f (x) = sinx.f (x) = cosx.f (x) = tanx.f (x) = (sinx)(cosx).f (x) = sin2 x.

Note: Remember that this notation means the square of sinx,i.e.,

sin2 x = (sinx)2.

f (x) = cos2 xf (x) = sinx+ cosx

Correct Answers:• O• E• O• O• E• E• N

13. (1 pt) 1060Library/set6 Trigonometry/s6p2.pgIn the next few problems all the answers are expressions of theform

asin(bx− c)+d or acos(bx− c)+dwhere the parameters a, b, c and d are simple numbers.

Often they are very simple, for example the choice

a = b = 1, c = d = 0

in the first expression just gives sinx. In any case, you are notasked to enter values of the parameters, your answers should bealgebraic expression involving the trigonometric functions.

The graphs may show up more clearly in a browser than on theprinted hard copy of this assignment.

The figure

shows the graph of the functionf (x) = .The figure

shows the graph of the functionf (x) = .

Correct Answers:

• sin(x)• cos(x)

14. (1 pt) 1060Library/set6 Trigonometry/s6p11.pgThe figure

shows the graph of the functionf (x) = .

Correct Answers:

• tan(2*x)

3

15. (1 pt) 1060Library/set7 Trigonometry/q3.pgYou are driving along the highway and see a sign indicating thatyou are about to enter a downhill slope of 10%. You realize thatyou are going to travel downward at an angle of degreesto the horizontal. (Enter your answer as a mathematical expres-sion or with at least two digits beyond the decimal point.)Hint: A descent of 10% means that you are going to descend by1 foot for every ten feet that you travel in a horizontal direction.The angle of your descent is the acute angle that the highwaymakes with a horizontal line. Draw a picture and use an inversetrig function.


16. (1 pt) 1060Library/set7 Trigonometry/q4.pgYou are about to land your Cessna airplane in Salt Lake City.You are approaching the runway at a ground speed of 72 milesper hour and you are sinking at 380 feet per minute. (The groundspeed is the speed of the point on the ground directly underneathyour plane. You can also think of it as the horizontal componentof your current velocity.) You are going to hit the runway at anangle of degrees. (Enter your answer as a mathe-matical expression, or with at least three digits beyond the deci-mal point.)Hint: Draw a triangle showing your descent and the horizontaldistance you cover in one minute.


17. (1 pt) 1060Library/set7 Trigonometry/s1.pgThis and the following three problems cover some subtleties ofthe inverse trig functions. In these problem, WeBWorK willnot accept as answers mathematical expressions that involvethe trigonometric functions or their inverses. You need to en-ter numbers.

In this problem let’s measure angles in degrees. As discussed inclass, the inverse sin function returns an angle between -90 and+90 degrees. This gives perhaps surprising results when youevaluate the sine somewhere and then the inverse sine at the re-sult. You can just key the problems below and in the followingproblems into your calculator, but try to get them without a cal-culator. If you yield to temptation and do use your calculator, atleast think about the answers when they are unexpected.Complete the following equations:arcsin(sin(37◦)) = ◦,arcsin(sin(−25◦)) = ◦,arcsin(sin(100◦)) = ◦,

Correct Answers:• 37• -25• 80

18. (1 pt) 1060Library/set7 Trigonometry/s2.pgThis is like the previous problem, except that angles are mea-sured in radians.Complete the following equations:arcsin(sin(1)) = ,arcsin(sin(−1.1)) = ,arcsin(sin(2)) = .

Correct Answers:• 1• -1.1• 1.14159265358979

19. (1 pt) 1060Library/set7 Trigonometry/s3.pg

Complete the following equations:arccos(cos(37◦)) = ◦,arccos(cos(−25◦)) = ◦,arccos(cos(100◦)) = ◦,


20. (1 pt) 1060Library/set8 Trigonometry/s3.pgThe Great Pyramid of Cheops has a square base with a lengthof 756 feet. Its height is 482 feet. If you walk straight up fromthe center of the north side to the top of the pyramid you haveto climb an angle of degrees.You decide to simplify your life and walk up along one of theridges. Thus you have to climb only at an angle of de-grees.On your way up the ridge you walk a distance of feet.Hint: For the first two parts draw right triangles and use an in-verse trig function. For the third part just use the PythagoreanTheorem.

Correct Answers:• 51.8953094143326• 42.0395484835217• 719.786079331908

21. (1 pt) 1060Library/set8 Trigonometry/s8.pg

You are hiking along the west shore of a river that’s flowingdue north. You notice a tree on the far shore at a bearing of 30degrees. You walk on for another 100 feet and you are stoppedby an unclimbable cliff. You contemplate swimming across theriver and wonder how wide it is. The tree on the other sidenow appears at a bearing of 45 degrees. Remembering your trigclass, you figure out that the river is feet wide.Hint: Again, consider two right triangles. One of them isisosceles, which simplifies things.



4

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Prep 5.1-5.3 is due : 10/27/2008 at 12:00pm MDT.The






1. (13 pts) set9 Analytic Trigonometry/s1.pg

Do not use a calculator to do this problem. You should beable to do it by using properties of the trig functions.

Suppose

sin(u) =35

and cos(u) is positive. Then

cos(u) =tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin(u+ π

2 ) =cos(u+ π

2 ) =tan(u+ π

2 ) =Correct Answers:

• 0.8• 0.75• -0.6• 0.8• -0.75• -0.6• -0.8• 0.75• 0.8• -0.6• -1.33333333333333

2. (13 pts) 1060Library/set9 Analytic Trigonometry/s4.pg

Suppose

cosu =513

and sinu is negative. Here are some small variations on theprevious problems:

sin(u) =sin(u−π) =cos(u−π) =sin

(u− π

2

)=

cos(u− π

2

)=

Correct Answers:

• -0.923076923076923• 0.923076923076923• -0.384615384615385• -0.384615384615385• -0.923076923076923

3. (13 pts) 1060Library/set9 Analytic Trigonometry/s6.pgThe subject of this home work set is trigonometric identities.WeBWorK has the usually useful feature that it will not dis-tinguish between any equivalent expressions. For example if Iasked you to express sin2 x in terms of cosx, and told ww thatthe correct answer is 1− cos2 x, then WeBWorK would be justas happy if you typed sin2x, and so bypassed the entire problem!

To finesse this feature we have to do something contrived. Ineach of the next few problems I will tell you to use a differentvariable for certain things.

Here is an easy warm up question and illustration:

Use C to denote cosx, and express sin2 x in terms of C:sin2(x) = .Since sin2 x+ cos2 x = 1, in this case you would of course enter

1−C2

1

as your answer. You can type it, for example, as 1-C**2 . If youfeel adventurous you can also type (1-C)(1+C), but you mustuse C instead of cosx!

Note that mathematics and WeBWorK are case sensitive , soyou must use upper case C in this problem. For the fun of it youmight check the cryptic error message you get when you use alower case c or some other variable!

Computers are fun!Let’s see if you got all this. Use S to denote sinx, and express

cos2 x in terms of S: cos2(x) = .Correct Answers:

• 1-C*C• 1-S*S


Use C to denote cosx, and express1+ tan2(x) = in terms of C.Similarly, express(1− sinx)(1+ sinx) = in terms of C.Hint: For the first part use the definition of the tangent function.For the second part use the third binomial formula:

(a+b)(a−b) = a2−b2.

Correct Answers:

• 1/(C*C)• C*C


Use C to denote cosx, and expresscos(x+20,000π) = in terms of C.Similarly, expresscos(x+20,001π) = in terms of C.

Correct Answers:

• C• -C

6. (13 pts) 1060Library/set9 Analytic Trigonometry/s10.pgUse T to denote tanx, and expresstan(x+20,000π) = in terms of T .Similarly, expresstan(x+20,001π) = in terms of T .

Correct Answers:

• T• T

7. (13 pts) 1060Library/set10 Analytic Trigonometry/e2.pgThe purpose of this problem is to help you prepare for Exam 2.Note that this will take place on Monday, March 31, i.e., twodays before this set closes. Work this problem well before theexam and let me know if you have any questions.

Remember that the exam is closed books and notes. You mayuse a calculator (and in fact I assume you have one available

that can evaluate trigonometric functions and their inverses). Toavoid distraction and confusion I won’t be able to answer ques-tions during the exam.

The exam will cover sections 4.7–5.4 of the textbook. Theproblems will be taken verbatim from the textbook. They willbe simple, and there will be one or two for each item in thefollowing list:

Using inverse trigonometric functions to find all angles witha certain value of a trigonometric function. For example,consider the problem of finding all angles 0≤ θ < 2π such that

cosθ =12.

We obtain one such angle by applying the inverse cos function:

θ = arccos12

=π

3.

Note that the function f (x) = cosx is symmetric about the verti-cal line x = π. Thus the other angle in the given interval with cosequal to one half is θ = 2π− π

3 = 5π

3 . You can solve problemslike this by finding one solution (using an inverse trig function)and then finding the others by looking at the graph of the orig-inal function or the unit circle. Try your hand on this problem:There are two angles 0◦ ≤ α < 360◦ such that

sinα =−√

22

.

The smaller is α = degrees and the larger is α =degrees.For some particular angles we know exact values of the trigono-metric functions, and these of course give rise to correspondstatements bout inverse trigonometric functions. For examplewe know that

cosπ

3=

12⇐⇒ arccos

12

=π

3.

Think about these relationships for the angles 0, π

6 , π

4 , π

3 , and π

2 .

We are considering only three trigonometric functions, sin, cos,and tan. The cos and sin are the coordinates of a point on theunit circle and tan is the ratio of sin and cos. Everything flowsfrom there. If for a given angle you know any two of sin, cos,and tan, you can compute the missing value. For example, sup-pose you know that

tanx =−√

52

andsinx =−√

53

.

Then cosx = .

A big subject in trigonometry is trigonometric identities.There are very many of them, too many to remember. How-ever, they are highly interdependent, and some are much more

2

important than others. Easily the two most important are

sin2 u+ cos2 u = 1

and

tanu =sinucosu

.

which flow directly from the definitions of the trigonometricfunctions. These two identities are so basic and used so fre-quently that by now they should be as familiar and natural toyou as the multiplication table. Some other identities like thoseexpressing the facts that the sin and tan are odd, the cos is even,adding pi to an angle changes the sign of sin and cos, but not thesin of tan, follow immediately by looking at the unit circle. Oth-ers, like the effects of shifting a graph by π

2 left or right can beeasily verified and derived if necessary by looking at the graphsof the trigonometric functions. I will assume that you know orcan derive these formulas. The third tier of identities, like thesum and difference formulas will be listed on the exam if theyare needed. Look at the exercises of section 5.2, and do thosethat only involve sin, cos, and tan.

Solving trigonometric equations. The basic principle of equa-tion solving holds for trigonometric equations just like anyother: To isolate the variable figure out what bothers you andthen get rid of it by doing the same thing on both sides of theequation. For trigonometric equations you need to apply inversetrig functions in some places. Also keep in mind that trigono-metric equations usually have many solutions, and make sureyou understand whether the questions asks for just one solution,all solutions in some interval, or all solutions.

For example, measuring angles in radians, the smallest positivenumber x such that

3sin(2x−1) = 1

is x = .

Graphing functions. We have worked long and frequently withthe graphs of the trigonometric functions. The inverse trig func-tions also have their graphs, and you should be familiar withthem. Make sure you understand that and how you have to re-duce the domain of or range of some functions to define theinverses of the trigonometric functions. There is a problem onthe test that involves the graph of a trig function and its inverse.The solutions show these graphs in black and white. Visit thesolutions of this problem after the set closes to see a coloredversion of those graphs.

A major application area of trigonometric functions is the ge-ometry of right triangles. There will be two very simple wordproblems taken directly from the textbook.

Correct Answers:• 225• 315• 0.666666666666667• 0.669918454727061

8. (13 pts) set10 Analytic Trigonometry/p1.pgIn the next three problems you are asked to solve basic trig equa-tions. You may use ”pi” for π. You may also use inverse trigfunctions in your answer, so there is no need to use a calculatorto evaluate decimal numbers.

The smallest positive number for which

tan(x) = 1is x = .

Enter arctan(1).The next larger such number is x = . We know that tan

is π-periodic, and we know from the shape of the graph that thenext time tangent will take the same value is in the next period,so enter

arctan(1)+ piThe smallest positive number for which

tan(3x) = 1

is x = . The next larger such number is x = .Hint: Apply an inverse trig function and use the periodicity ofthe tan function.

Correct Answers:• 0.785398163397448• 3.92699081698724• 0.261799387799149• 1.30899693899575

9. (13 pts) 1060Library/set10 Analytic Trigonometry/p2.pgThe smallest positive number for which

3tanx = 1

is x = . The next larger such number is x = .Hint: This is the like the preceding problem, except that youreverse the sequence of applying the inverse trig function anddividing by 3.

Correct Answers:• 0.321750554396642• 3.46334320798644

10. (13 pts) 1060Library/set10 Analytic Trigonometry/p3.pgThe smallest positive number for which

3sinx = 1

is x = . The next larger such number is x = .Hint: Divide by 3 and solve the resulting equation. Apply aninverse trig function and exploit the symmetry of the sin func-tion.

Correct Answers:• 0.339836909454122• 2.80175574413567

3


4

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Transformations of Trig Functions is due : 10/29/2008 at 03:00pm MDT.The







Letf (x) = 3sin(5x−4)+2.


Correct Answers:• 3• 1.25663706143592• 0.8• 2




Correct Answers:• 7• 0.666666666666667• 0.212206590789194• -3

3. (13 pts) 1060Library/set6 Trigonometry/q3.pgLet

f (x) = 12sin(4x−2)+3.


Correct Answers:• 12• 1.5707963267949

• 0.5• 3

4. (13 pts) set6 Trigonometry/s6p2.pgIn the next few problems all the answers are expressions of theform

asin(bx− c)+d or acos(bx− c)+d

where the parameters a, b, c and d are simple numbers.

Often they are very simple, for example the choice

a = b = 1, c = d = 0

in the first expression just gives sinx. In any case, you are notasked to enter values of the parameters, your answers should bealgebraic expression involving the trigonometric functions.

The graphs may show up more clearly in a browser than on theprinted hard copy of this assignment.

The figure{

image(”p2a.gif”, height =¿ 200, width =¿ 900);} shows the graph of the function

f (x) = . Hint: The figure has x-intercepts at 0 and π.

The figure{

image(”p2b.gif”, height =¿ 200, width =¿ 900);} shows the graph of the function

f (x) = . Hint: The figure has an x-intercept at π

2 .

Correct Answers:

• sin(x)• cos(x)

1

5. (13 pts) set6 Trigonometry/s6p11.pgThe figure{

image(”p11.gif”, height =¿ 600, width =¿ 600);} shows the graph of the function

f (x) = . Hint: The figure has x-intercepts at 0, π

2 , π...Correct Answers:

• tan(2*x)



f (x) = . Hint: The figure has x-intercepts at π

2 , 3π

2 ...The figure{


f (x) = . Hint: The figure has x-intercepts at π, 2π...Correct Answers:

• -cos(x)• -sin(x)




2 , π...The figure{


f (x) = . Hint: The figure has x-intercepts at 0, 1, 2, 3....Correct Answers:

• sin(2x)

• sin(3.14159265358979*x)



f (x) = . Hint: f (π) = 0 and the maximum value of f is2.The figure{


g(x) = . Hint: g(π) = 0 and the minimum value of g is-3.

Correct Answers:• 2*sin(x)• 3*sin(x)



f (x) = . Hint: f achieves its maximum value of 2 at π

2and f is non-negative.

Correct Answers:• sin(x)+1



f (x) = . Hint: f achieves its maximum value of 2 at 0. fachieves its minimum value of 0 at 1.

Correct Answers:• cos(3.14159265358979*x)+1


2

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Exam3practice is due : 11/03/2008 at 02:19pm MST.The






1. (1 pt) 1060Library/set9 Analytic Trigonometry/s1.pg

Suppose

sin(u) =35

and cos(u) is positive. Then

cos(u) =tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin(u+ π

2 ) =cos(u+ π

2 ) =tan(u+ π

2 ) =Correct Answers:

• 0.8• 0.75• -0.6• 0.8• -0.75• -0.6• -0.8• 0.75• 0.8• -0.6• -1.33333333333333


Suppose

sin(u) =35

and cos(u) is negative. Then

cos(u) =tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin

(u+ π

2

)=

cos(u+ π

2

)=

tan(u+ π

2

)=

Correct Answers:

• -0.8• -0.75• -0.6• -0.8• 0.75• -0.6• 0.8• -0.75• -0.8• -0.6• 1.33333333333333


In this problem you need to get everything right to get credit.The idea is that you go over everything an think about the wholecontext. Use the preceding problem as a guide.

Suppose

sin(u) =1213

and cos(u) is negative. Note that cosu is a rational number. Irecommend you enter fractions below.

cos(u) =1

tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin

(u+ π

2

)=

cos(u+ π

2

)=

tan(u+ π

2

)=

Correct Answers:• -0.384615384615385• -2.4• -0.923076923076923• -0.384615384615385• 2.4• -0.923076923076923• 0.384615384615385• -2.4• -0.384615384615385• -0.923076923076923• 0.416666666666667


Suppose

cosu =513

and sinu is negative. Here are some small variations on theprevious problems:

sin(u) =sin(u−π) =cos(u−π) =sin

(u− π

2

)=

cos(u− π

2

)=

Correct Answers:• -0.923076923076923• 0.923076923076923• -0.384615384615385• -0.384615384615385• -0.923076923076923


This is like the preceding problem, except that you need toget all answers correctly to get credit.

Suppose

cosu =35

and sinu is positive.

sin(u) =sin(u−π) =cos(u−π) =

sin(u− π

2

)=

cos(u− π

2

)=

Correct Answers:

• 0.8• -0.8• -0.6• -0.6• 0.8

6. (1 pt) 1060Library/set9 Analytic Trigonometry/s6.pgThe subject of this home work set is trigonometric identities.WeBWorK has the usually useful feature that it will not dis-tinguish between any equivalent expressions. For example if Iasked you to express sin2 x in terms of cosx, and told ww thatthe correct answer is 1− cos2 x, then WeBWorK would be justas happy if you typed sin2x, and so bypassed the entire problem!

To finesse this feature we have to do something contrived. Ineach of the next few problems I will tell you to use a differentvariable for certain things.

Here is an easy warm up question and illustration:

Use C to denote cosx, and express sin2 x in terms of C:sin2(x) = .Since sin2 x+ cos2 x = 1, in this case you would of course enter

1−C2

as your answer. You can type it, for example, as 1-C**2 . If youfeel adventurous you can also type (1-C)(1+C), but you mustuse C instead of cosx!

Note that mathematics and WeBWorK are case sensitive , soyou must use upper case C in this problem. For the fun of it youmight check the cryptic error message you get when you use alower case c or some other variable!

Computers are fun!Let’s see if you got all this. Use S to denote sinx, and express

cos2 x in terms of S: cos2(x) = .Correct Answers:

• 1-C*C• 1-S*S


Use C to denote cosx, and express1+ tan2(x) = in terms of C.Similarly, express(1− sinx)(1+ sinx) = in terms of C.Hint: For the first part use the definition of the tangent function.

2

For the second part use the third binomial formula:

(a+b)(a−b) = a2−b2.

Correct Answers:

• 1/(C*C)• C*C


Use C to denote cosx, and expresscos(x+20,000π) = in terms of C.Similarly, expresscos(x+20,001π) = in terms of C.

Correct Answers:

• C• -C

9. (1 pt) 1060Library/set9 Analytic Trigonometry/s10.pgUse T to denote tanx, and expresstan(x+20,000π) = in terms of T .Similarly, expresstan(x+20,001π) = in terms of T .

Correct Answers:

• T• T

10. (1 pt) 1060Library/set10 Analytic Trigonometry/p1.pgThe smallest positive number for which

tan(3x) = 1

is x = . The next larger such number is x = .Hint: Apply an inverse trig function and use the periodicity ofthe tan function.

Correct Answers:

• 0.261799387799149• 1.30899693899575


3tanx = 1

is x = . The next larger such number is x = .Hint: This is the like the preceding problem, except that youreverse the sequence of applying the inverse trig function anddividing by 3.

Correct Answers:

• 0.321750554396642• 3.46334320798644


3sinx = 1

is x = . The next larger such number is x = .Hint: Divide by 3 and solve the resulting equation. Apply aninverse trig function and exploit the symmetry of the sin func-tion.

Correct Answers:

• 0.339836909454122• 2.80175574413567


3sin(2x−1) = 1

is x = .Hint: Isolate the sin, apply an inverse trig function, and solvethe resulting linear equation.

Correct Answers:

• 0.669918454727061


3sin(2x−6) = 1

is x = .Hint: Proceed as in the preceding problem to find some solutionof the equation. Then look for the smallest positive.

Correct Answers:

• 0.0283258011372678

15. (1 pt) 1060Library/set10 Analytic Trigonometry/p6.pg


3sin(2x−5) = 1

is x = .Hint: This is much like the preceding problem.

Correct Answers:

• 0.759285218478043

16. (1 pt) 1060Library/set10 Analytic Trigonometry/p7.pg


4cos2 x−9cosx+2 = 0

is x = .Hint: Solve the quadratic equation for cosx and then solve forx.

Correct Answers:

• 1.31811607165282

3

17. (1 pt) set10 Analytic Trigonometry/p10.pgExpress sin(3t) in terms of sin t and cos t. Use the folowing for-mulas:

sin(u+ v) = sin(u)cos(v)+ cos(u)sin(v)

cos(u+ v) = cos(u)cos(v)− sin(u)sin(v)

Hint: 3t = 2t + t and 2t = t + t.Enter the resulting expression for sin(3t) here

,using upper case S to denote sint and upper case C to denotecos t. For example, if your answer was 3sin t cos t you’d simplyenter 3*S*C . Remember that WeBWorK (and mathematics) iscase sensitive! You may be able to find an equivalent expressionfor sin(3t) in the literature, but WeBWorK will not recognize theequivalence. You have to follow the procedure outlined above.

Hint: Apply suitable sum formulas twice.Correct Answers:

• 3*S*C**2-S**3

18. (1 pt) set11 Analytic Trigonometry/p1.pgThe smallest positive number for which

sin(2x)− sinx = 0

is x = .Use the fact that

sin(2u) = 2sin(u)cos(u)

Hint: Apply a multiple angle or difference to product rule.Correct Answers:

• 1.0471975511966

19. (1 pt) set11 Analytic Trigonometry/p4.pgThe smallest positive number for which

sin t− cos(2t) = 0

is t = .Use the fact that

cos(2u) = cos2(u)− sin2(u)

Then use the Pythagorean identity.Hint: This is like the preceding problem.

Correct Answers:

• 0.523598775598299


Letf (x) = 3sin(5x−4)+2.


Correct Answers:• 3• 1.25663706143592• 0.8• 2




Correct Answers:• 7• 0.666666666666667• 0.212206590789194• -3

22. (1 pt) 1060Library/set6 Trigonometry/q3.pgLet

f (x) = 12sin(4x−2)+3.


Correct Answers:• 12• 1.5707963267949• 0.5• 3

23. (1 pt) set6 Trigonometry/s6p3.pgThe figure{


f (x) = . Hint: The figure has x-intercepts at π

2 , 3π

2 ...The figure{


f (x) = . Hint: The figure has x-intercepts at π, 2π...Correct Answers:

• -cos(x)• -sin(x)

4




2 , π...The figure{


f (x) = . Hint: The figure has x-intercepts at 0, 1, 2, 3....Correct Answers:

• sin(2x)• sin(3.14159265358979*x)



f (x) = . Hint: f (π) = 0 and the maximum value of f is2.The figure{


g(x) = . Hint: g(π) = 0 and the minimum value of g is-3.

Correct Answers:

• 2*sin(x)• 3*sin(x)



f (x) = . Hint: f achieves its maximum value of 2 at π

2and f is non-negative.

Correct Answers:• sin(x)+1



f (x) = . Hint: f achieves its maximum value of 2 at 0. fachieves its minimum value of 0 at 1.

Correct Answers:• cos(3.14159265358979*x)+1




2 , π...Correct Answers:

• tan(2*x)


5

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Law of Sines is due : 11/19/2008 at 03:00pm MST.The






1. (15 pts) 1060Library/set11 Analytic Trigonometry/p5.pgIn this and the next few problems we consider general (not nec-essarily or usually right) triangles and use the notation describedon p.430 of the textbook, and indicated in this Figure:

The angles are labeled A, B, and C, and the sides opposite theseangles have lengths a, b, and c, respectively.Suppose you are given a triangle where:

A = 38◦, B = 39◦ and c = 37.

ThenC = ,a = , andb = .Note: In this problem angles are measured in degrees.

Correct Answers:

• 103• 23.3786683420639• 23.8973417907459

2. (15 pts) 1060Library/set11 Analytic Trigonometry/p6.pgSuppose you are given a triangle where:

A = 0.9, B = 0.6 and c = 9.

ThenC = ,a = , andb = .Note: In this problem angles are measured in radians.

Correct Answers:• 1.64159265358979• 7.06764673639984• 5.09454416192718

3. (15 pts) 1060Library/set11 Analytic Trigonometry/p7.pgSuppose you are given a triangle where:

A = 30◦, a = 12, and b = 10.

ThenB = ,C = , andc = .Note: In this problem angles are measured in degrees.

Correct Answers:• 24.6243183521641• 125.375681647836• 19.5689661524801

4. (15 pts) 1060Library/set11 Analytic Trigonometry/p8.pgThere are two triangles for which

A = 30◦, a = 7, and b = 10.

The larger one hasc = , and the smaller one hasc = .

Correct Answers:• 13.5592335234107• 3.76127455227803

1

5. (15 pts) 1060Library/set11 Analytic Trigonometry/p9.pgFor each of the following combinations of A, a, and b indicateif there are 0, 1, or 2 triangles with these data. (Enter the digit0, 1, or 2.)

A = 30◦, b = 10, a = 12 :A = 30◦, b = 10, a = 8 :A = 30◦, b = 10, a = 3 :


6. (15 pts) 1060Library/set11 Analytic Trigonometry/p12.pg(This is problem 38 on page 500 of the textbook.) A flagpole atright angle to the horizontal is located on a slope that makes anangle of 12◦ with the horizontal. The pole’s shadow is 16 me-ters long and points directly up the slope. The angle of elevationfrom the tip of the shadow to the sun is 20◦. The height of thepole is meters.Hint: Draw a triangle and apply the Law of Sines. Note thatthe shadow points up the slope starting from the bottom of theflag pole. The angle of elevation is the angle made with thehorizontal (not with the ground).


7. (15 pts) 1060Library/set11 Analytic Trigonometry/p13.pg(This is a slight modification of problem 37 on page 437 of thetextbook.) A flagpole at right angle to the horizontal is locatedon a slope that makes an angle of 12◦ with the horizontal. Thepole’s shadow is 16 meters long and points directly down theslope. The angle of elevation from the tip of the shadow to thesun is 20◦. The height of the pole is meters.

Hint: Draw a triangle and apply the Law of Sines. Note thatthe shadow points down the slope starting from the bottom ofthe flag pole. The angle of elevation is the angle made with thehorizontal (not with the ground).


8. (15 pts) 1060Library/set12 Additional Topics in Trigonometry-/p3.pgSuppose you are given a triangle with

A = 60◦, a = 5, b = 3.

ThenB = degrees,C = degrees, andc = .Enter your answers with two digits beyond the decimal point.

Correct Answers:• 31.3064462486731• 88.6935537513269• 5.77200187265877


A = 40◦, B = 37◦, c = 5.

Thena = , andb = .Enter your answers with at least 3 digits beyond the decimalpoint.

Correct Answers:• 3.29847782129995• 3.08822615196361


2

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Prep Dot Product is due : 11/24/2008 at 12:00pm MST.The


This material is in section 6.4




1. (15 pts) setPrep Dot Product/prob1.pgu = 〈−1,3〉 and v = 〈2,−4〉 Compute

u ·v

Correct Answers:

• -14

2. (15 pts) setPrep Dot Product/prob2.pgCompute

〈−1,1〉 · 〈2,3〉

Correct Answers:

• 1


〈2,1〉 · 〈8,3〉

Correct Answers:

• 19


〈10,1〉 · 〈0,3〉

Correct Answers:

• 3


|| 〈10,1〉 ||


6. (15 pts) setPrep Dot Product/prob6.pg

u = 〈2,3〉Compute

||u||




||u||




||u||



1

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Law of Cosines is due : 11/26/2008 at 03:00pm MST.The






1. (13 pts) 1060Library/set12 Additional Topics in Trigonometry-/e3.pgThe purpose of this problem is to help you prepare for exam3. Its focus will be on solving triangles. A triangle has threesides and three angles. The awkward phrase ”solving a trian-gle” means given three of those data, find the other three. (Itmay also include finding other quantities, like finding some orall of the heights, or finding the area.)In this problem, as usual, the angles are labeled A, B, and C,and the sides opposite these angles have lengths a, b, and c, re-spectively. In abstract problems the lengths aren’t specified inany particular units, but we assume that the units are same forall lengths involved. We can distinguish a number of cases forthe three given data. For each cases we proceed differently. Thecases are:

abc. Three sides and no angles are given. In that case any twosides have to add up to more than the third side. (For example,there is no triangle for which two sides are one foot long, andthe third has a length of one mile.) If so there is a unique tri-angle with those three sides. You can find the angles using theLaw of Cosines. For example, suppose

a = 3, b = 4, c = 6.

ThenB = degrees. (You can compute A and C similarly.)

Axy One angle and two sides are given. There are two subcases:

abC The two sides make up the angle. There is a unique trian-gle. Find the opposite side using the law of cosines. Then findthe other angles using the either the law of sines or the law of

cosines. For example, suppose

a = 3, b = 4, C = 40◦.

Then c = .

abA One of the sides is opposite the angle. Apply the law ofsines. There are several subcases:

a is too small, there is no solution.a is just large enough so that there is a unique right triangle.a < b but it’s large enough so that there are two triangles.a = b There is one isosceles triangle.a > b There is a unique triangle.

For example, suppose A = 40◦ and b = 5. Then there will beno triangle if a < . If in fact a equals that critical valuethen c = .

If a is greater than that critical value, but less than 5, for ex-ample, if a = 4, then there are two triangles, with two possiblevalues of B. The smaller of these two values is

B = degreesand the larger is degrees.

ABx. If we know two angles then we know all three angles.Given two angles and one side we can use the Law of Sines tofind all sides, and there is unique angle. Of course the given twoangles need to add to less than 180 degrees, otherwise there isno triangle. For example, if

A = 30◦, B = 80◦, c = 3,

then a= .

ABC. If we only know three angles (adding to 180 degrees) and1

no sides then there are infinitely many triangles with those an-gles. Those triangles are called similar . They have the propertythat ratios of corresponding sides are equal.

On the exam there will be a problem for each of the abovecases, and one simple word problem involving the solution of atriangle.

Correct Answers:

• 36.3360575146139• 2.57195127581075• 3.2139380484327• 3.83022221559489• 53.4641490143885• 126.535850985612• 1.59626665871387

2. (13 pts) 1060Library/set12 Additional Topics in Trigonometry-/p1.pgIn this and the following problems we consider general (not nec-essarily or usually right) triangles and use the notation describedon p. 494 of the textbook, and indicated in this Figure:

The angles are labeled A, B, and C, and the sides opposite theseangles have lengths a, b, and c, respectively. In abstract prob-lems the lengths aren’t specified in any particular units, but weassume that the units are same for all lengths involved.Suppose you are given a triangle with

a = 4, b = 7, c = 10.

ThenA = degrees,B = degrees, andC = degrees.Enter your answers with two digits beyond the decimal point.

Correct Answers:

• 18.1948723387668• 33.1229402077438• 128.682187453489


A = 60◦, b = 6, c = 9.

Thena = ,B = degrees, andC = degrees.Enter your answers with two digits beyond the decimal point.

Correct Answers:• 7.93725393319377• 40.8933946491309• 79.1066053508691


A = 60◦, a = 5, b = 3.


Correct Answers:• 31.3064462486731• 88.6935537513269• 5.77200187265877


a = 8, b = 3, c = 8.


Correct Answers:• 79.1930771251397• 21.6138457497207• 79.1930771251397


A = 22◦, B = 51◦, c = 5.



2

• 4.06327562741417

7. (13 pts) 1060Library/set12 Additional Topics in Trigonometry-/p6.pgYou leave your friend behind on the shore and you travel 3 milesdue east in your boat. Then you travel 2 miles northeast. Thenyou travel 1 mile due north. Your friend can see you at a distanceof miles

and at a bearing of degrees.

Enter your answers with at least 3 digits beyond the decimalpoint.

Hint: Carefully draw a picture. Use the Laws of Sines andCosines.

Correct Answers:• 5.03127304953575• 61.3249499368952

8. (13 pts) 1060Library/set12 Additional Topics in Trigonometry-/p7.pgOrienteering is a sport in which competitors use a map and acompass to find their way through unfamiliar terrain. Your taskto is to travel along a triangular course. The first leg is due west,and 6,000 meters long. The other two legs are north of the firstleg. The second leg is 4,000 m long, and the third leg is 5,000meters long. Your compass bearing (the direction in which youare moving) during the second leg is degrees

During the third leg of your course your bearing isdegrees.


Hint: Carefully draw a picture. Use the Law of Cosines.Correct Answers:

• 34.2288663278126• 131.409622109271

9. (13 pts) 1060Library/set12 Additional Topics in Trigonometry-/p10.pg

Calculating the necessary aircraft heading to counter a windvelocity and proceed along a desired bearing to a destination isa classic problem in aircraft navigation. It makes good use ofthe law of sines and the law of cosines.

Suppose you wish to fly in a certain direction relative to theground. The wind is blowing at 50mph at an angle of 40 degrees

to that direction. Your plane is flying at 100mph with respect tothe surrounding air. The situation is illustrated in this Figure(where your desired direction of travel is due East):

Then you head into the wind at an angle of

degrees (enter your value of α), and your ground speedis miles per hour (enter your value of x).Hint: Apply the Law of Sines to get the angle α and the law ofcosines to get the ground speed.

Correct Answers:

• 18.7472372510375• 132.996784946447

10. (13 pts) 1060Library/set12 Additional Topics in Trigonometry-/p11.pgThis is like the preceding problem, except it’s more general.Suppose the speed of the wind is w, the speed of the plane isp, and the ground speed is g. Let A denote the angle you planeneeds to make with the wind, and B the angle that the windmakes with your desired direction of travel, as illustrated in thisFigure:

For simplicity assume that the angles are measured in ra-dians so that you can use the WeBWorK trig functions andtheir inverses without conversion.

ThenA = .

Your answer will be a mathematical expression in B, p and w.Once you have the angle A you can compute your ground speedusing either the Law of Sines or the Law of Cosines.Hint: Apply the Law of Sines to get the angle.

Correct Answers:

• asin(w*sin(B)/p)


3

Hsiang-Ping Huang math1060fall2008-1WeBWorK assignment number Exam 4 practice is due : 11/30/2008 at 06:44pm MST.The






1. (1 pt) 1060Library/set12 Additional Topics in Trigonometry/e3.pgThe purpose of this problem is to help you prepare for exam3. Its focus will be on solving triangles. A triangle has threesides and three angles. The awkward phrase ”solving a trian-gle” means given three of those data, find the other three. (Itmay also include finding other quantities, like finding some orall of the heights, or finding the area.)In this problem, as usual, the angles are labeled A, B, and C,and the sides opposite these angles have lengths a, b, and c, re-spectively. In abstract problems the lengths aren’t specified inany particular units, but we assume that the units are same forall lengths involved. We can distinguish a number of cases forthe three given data. For each cases we proceed differently. Thecases are:

abc. Three sides and no angles are given. In that case any twosides have to add up to more than the third side. (For example,there is no triangle for which two sides are one foot long, andthe third has a length of one mile.) If so there is a unique tri-angle with those three sides. You can find the angles using theLaw of Cosines. For example, suppose

a = 3, b = 4, c = 6.

ThenB = degrees. (You can compute A and C similarly.)

Axy One angle and two sides are given. There are two subcases:

abC The two sides make up the angle. There is a unique trian-gle. Find the opposite side using the law of cosines. Then findthe other angles using the either the law of sines or the law of

cosines. For example, suppose

a = 3, b = 4, C = 40◦.

Then c = .

abA One of the sides is opposite the angle. Apply the law ofsines. There are several subcases:

a is too small, there is no solution.a is just large enough so that there is a unique right triangle.a < b but it’s large enough so that there are two triangles.a = b There is one isosceles triangle.a > b There is a unique triangle.

For example, suppose A = 40◦ and b = 5. Then there will beno triangle if a < . If in fact a equals that critical valuethen c = .

If a is greater than that critical value, but less than 5, for ex-ample, if a = 4, then there are two triangles, with two possiblevalues of B. The smaller of these two values is

B = degreesand the larger is degrees.

ABx. If we know two angles then we know all three angles.Given two angles and one side we can use the Law of Sines tofind all sides, and there is unique angle. Of course the given twoangles need to add to less than 180 degrees, otherwise there isno triangle. For example, if

A = 30◦, B = 80◦, c = 3,

then a= .

ABC. If we only know three angles (adding to 180 degrees) and1

no sides then there are infinitely many triangles with those an-gles. Those triangles are called similar . They have the propertythat ratios of corresponding sides are equal.

On the exam there will be a problem for each of the abovecases, and one simple word problem involving the solution of atriangle.

Correct Answers:

• 36.3360575146139• 2.57195127581075• 3.2139380484327• 3.83022221559489• 53.4641490143885• 126.535850985612• 1.59626665871387

2. (1 pt) 1060Library/set12 Additional Topics in Trigonometry/p1.pgIn this and the following problems we consider general (not nec-essarily or usually right) triangles and use the notation describedon p. 494 of the textbook, and indicated in this Figure:

The angles are labeled A, B, and C, and the sides opposite theseangles have lengths a, b, and c, respectively. In abstract prob-lems the lengths aren’t specified in any particular units, but weassume that the units are same for all lengths involved.Suppose you are given a triangle with

a = 4, b = 7, c = 10.


Correct Answers:

• 18.1948723387668• 33.1229402077438• 128.682187453489

3. (1 pt) 1060Library/set12 Additional Topics in Trigonometry/p2.pgSuppose you are given a triangle with

A = 60◦, b = 6, c = 9.

Thena = ,B = degrees, andC = degrees.Enter your answers with two digits beyond the decimal point.

Correct Answers:

• 7.93725393319377• 40.8933946491309• 79.1066053508691


A = 60◦, a = 5, b = 3.


Correct Answers:

• 31.3064462486731• 88.6935537513269• 5.77200187265877


a = 8, b = 3, c = 8.


Correct Answers:

• 79.1930771251397• 21.6138457497207• 79.1930771251397


A = 44◦, B = 21◦, c = 5.


Correct Answers:

• 3.83235353593462• 1.97707641196074

2

7. (1 pt) 1060Library/set12 Additional Topics in Trigonometry/p6.pgYou leave your friend behind on the shore and you travel 3 milesdue east in your boat. Then you travel 2 miles northeast. Thenyou travel 1 mile due north. Your friend can see you at a distanceof miles

and at a bearing of degrees.


Hint: Carefully draw a picture. Use the Laws of Sines andCosines.

Correct Answers:

• 5.03127304953575• 61.3249499368952

8. (1 pt) 1060Library/set12 Additional Topics in Trigonometry/p7.pgOrienteering is a sport in which competitors use a map and acompass to find their way through unfamiliar terrain. Your taskto is to travel along a triangular course. The first leg is due west,and 6,000 meters long. The other two legs are north of the firstleg. The second leg is 4,000 m long, and the third leg is 5,000meters long. Your compass bearing (the direction in which youare moving) during the second leg is degrees

During the third leg of your course your bearing isdegrees.


Hint: Carefully draw a picture. Use the Law of Cosines.Correct Answers:

• 34.2288663278126• 131.409622109271

9. (1 pt) 1060Library/set12 Additional Topics in Trigonometry/p10.pg

Calculating the necessary aircraft heading to counter a windvelocity and proceed along a desired bearing to a destination isa classic problem in aircraft navigation. It makes good use ofthe law of sines and the law of cosines.

Suppose you wish to fly in a certain direction relative to theground. The wind is blowing at 50mph at an angle of 40 degreesto that direction. Your plane is flying at 100mph with respect tothe surrounding air. The situation is illustrated in this Figure(where your desired direction of travel is due East):

Then you head into the wind at an angle of

degrees (enter your value of α), and your ground speedis miles per hour (enter your value of x).Hint: Apply the Law of Sines to get the angle α and the law ofcosines to get the ground speed.

Correct Answers:• 18.7472372510375• 132.996784946447

10. (1 pt) 1060Library/set12 Additional Topics in Trigonometry-/p11.pgThis is like the preceding problem, except it’s more general.Suppose the speed of the wind is w, the speed of the plane isp, and the ground speed is g. Let A denote the angle you planeneeds to make with the wind, and B the angle that the windmakes with your desired direction of travel, as illustrated in thisFigure:

For simplicity assume that the angles are measured in ra-dians so that you can use the WeBWorK trig functions andtheir inverses without conversion.

ThenA = .

Your answer will be a mathematical expression in B, p and w.Once you have the angle A you can compute your ground speedusing either the Law of Sines or the Law of Cosines.Hint: Apply the Law of Sines to get the angle.

Correct Answers:• asin(w*sin(B)/p)

11. (1 pt) 1060Library/set13 Additional Topics in Trigonometry-/p14.pgLet

u =< 1,−1 > and v =< 1,2 > .

Thenu+ v =< , >,u− v =< , >,−3v =< , >,u · v = , and‖v‖= .

Correct Answers:• 2• 1• 0• -3• -3• -6• -1• 2.23606797749979

3

12. (1 pt) 1060Library/set13 Additional Topics in Trigonometry-/p16.pgYou are planning a ramp on which to pull rocks that weigh upto 2000 pounds. You can pull with a force of 100 pounds. Themaximum angle of elevation of that ramp is degrees. (Ig-nore friction. Still, it’s going to be a long ramp!)


13. (1 pt) setPrep Dot Product/prob1.pgu = 〈−1,3〉 and v = 〈2,−4〉 Compute

u ·v

Correct Answers:• -14


〈−1,1〉 · 〈2,3〉



〈2,1〉 · 〈8,3〉



〈10,1〉 · 〈0,3〉



|| 〈10,1〉 ||

Correct Answers:

• 10.0498756211209



||u||

Correct Answers:

• 3.60555127546399



||u||

Correct Answers:

• 1



||u||

Correct Answers:

• 5.3851648071345


4

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