Algebra I, Algebra II, and Transcendentals Review

© Department of Mathematical Sciences

George Mason University

Revised September 2017

Contents

Introduction ........................................................................................... 1

Courses Requiring the Placement Test ............................................... 2

Policies ................................................................................................... 2

Test Format ............................................................................................ 3

Learning Resources ............................................................................... 4

Basic Algebra.......................................................................................... 5

Fractions ................................................................................... 6

Decimal Notation ...................................................................... 7

Percents ..................................................................................... 8

Ratios…. .................................................................................... 10

Algebraic Expressions .............................................................. 11

Linear Equations in One Variable ............................................ 11

Linear Equations in Two Variables .......................................... 12

Graph/Slope of a line ................................................................ 13

Algebra I ................................................................................................. 14

Linear Inequalities in 1 Variable .............................................. 14

Absolute Value ......................................................................... 14

Systems of Linear Equations .................................................... 15

Linear Inequalities in 2 Variables ............................................. 16

Multiplying Algebraic Expressions .......................................... 16

Factoring Algebraic Expressions .............................................. 17

Rational Expressions ................................................................ 17

More on Exponents ................................................................... 18

Radicals .................................................................................... 19

Quadratic Equations ................................................................. 19

Functions Notation .................................................................... 20

Equations with Rational Expressions ....................................... 21

Algebra II……………………………………………………...……… 22

More on Functions .................................................................... 22

Graphs of Functions................................................................... 24

Transcendentals ..................................................................................... 25

Exponential Functions ............................................................... 25

Logarithmic Functions .............................................................. 26

Angle Measure .......................................................................... 26

Right Triangle and Circular Functions Definitions ................... 27

Graphs of Trigonometric Functions .......................................... 28

Inverse Trigonometric Functions .............................................. 29

Trigonometric Formulas and Identities ..................................... 30

Answers .................................................................................................. 31

1

Courses that Require the Placement Test

The Math Placement Test is given by the Department of Mathematical Sciences

to determine the readiness for the following courses:

Math 104 Trigonometry and Transcendental Functions and Math 105

Pre-Calculus

Math 108 Introductory Calculus with Business Applications

Math 113 Analytic Geometry and Calculus I

Math 123 Calculus Algebra/Trigonometry A

Math 125 Discrete Mathematics

CS 112 Computer Science I

Students should talk to an academic advisor to determine which Math course(s)

they are required to take. All students admitted to the university are advised to

take the Math Placement Test during the orientation process.

Policies

For current policies and required minimum scores please visit the

webpage:

http://math.gmu.edu/placement_test.php

2

Basic Algebra

Basic Operations

Example: John’s car insurance premium is $1515.00. He has the option of

making four equal payments but will be charged an additional $25 processing

fee. If he chooses the four-payment option, what is the amount of each payment?

Solution: The processing fee is added to the premium

00.1540$

00.25

00.1515$

This amount is then divided by 4 385

15404

Each payment will be $385.00

Exercise 1: A shipment of 1344 CD’s is to be packed into cartons containing 24

CD’s each. How many cartons are necessary to ship these CD’s?

Exercise 2: A loan of $10,020.00 is to be paid off in 60 equal monthly payments.

How much is each payment (excluding interest)?

Exercise 3: Susan’s Cell phone provider charges her $35.00 per month for the

first 300 minutes of calls and $0.70 for each minute over 300. If she uses 340

minutes of calls in a given month, how much will she be charged?

3

Fractions

A fraction or rational number is of the form q

p where p and q are integers, and

0q

p is called the numerator and q is called the denominator. The fraction is said to

be reduced to lowest terms if p and q have no factors (other than 1) in common.

The same number can be expressed as a fraction in many ways. These fractions

are called equivalent fractions: 4

1,

12

3,

20

5,

40

10,

48

12

Only the first fraction is in reduced form.

► To change from a mixed number to an improper fraction, multiply the whole

number by the denominator of the fraction and add the result to the numerator of

the fraction. This gives the numerator of the improper fraction.

Example: 3

5

3

21 ( 2315 )

8

43

8

35 ( 38543 )

► The product of two fractions d

c

b

a is

bd

ac. That is, multiply numerators to

get the numerator of the product and multiply denominators to get the

denominator of the product.

Examples: 99

8

9

4

11

2

5

12

5

2

1

6

5

26

► The quotient of two fraction d

c

b

a is equal to

c

d

b

a or

cb

ad

Example: 49 ounces of a solution is to be poured into test tubes with capacity

2

13 ounces. How many test tubes will be filled?

Solution: 147

249

2

749

2

1349 14 test tubes will be filled.

4

Exercise 4: Perform the indicated operation and give the result in reduced form.

a. 5

1

3

1 b. 6

8

3 c.

4

32

5

21 d.

8

5

3

26

► The sum of two fractions with a common denominator is c

ba

c

b

c

a

If fractions do not have the same denominator then they must be changed to an

equivalent form.

Example: Find the sum 8

5

3

11

Change each fraction to an equivalent form

24

32

8

8

3

4

3

11

24

15

3

3

8

5

8

5

Then add the equivalent fractions to get 124

47

24

15

24

32

8

5

3

1 or

24

231

Exercise 5: Perform the indicated operation and give the result in reduced

form

a. 8

7

4

16 b.

4

3

10

7 c.

6

1

8

5 d.

12

7

4

12

Decimal Notation

Rational numbers or fractions can also be expressed in decimal notation.

100

137 can be written as 7.13

.125 is equivalent to 1000

125 or

8

1

► When adding or subtracting numbers in decimal notation, make sure to align

the decimal point.

5

Example: Find the sum 78.31.23

Solution:

88.26

78.3

1.23

► When multiplying two numbers in decimal notation, first multiply the

numbers disregarding the decimal point(s). Next put the decimal point in the

product. The number of places to the right of the decimal point in the product

should be the SUM of the places to the right of the decimal point in each of the

factors.

Example: Find the product 1.235.6

Solution:

13335

12700

635

21

635

The product should have three places to the right of the decimal point.

The result will be 13.335

Example: Carlos is getting paid $10.80 per hour for the first 40 hours he is

working and one time and a half that wage for each extra hour. If he works 43.5

hours this week, how much will he earn?

Solution: Carlos is paid 20.16$5.180.10$ per hour of overtime

He will earn: 70.488)20.16$5.3()80.10$40(

Percents

The symbol % is used to indicate the fraction 100

1 or the ratio: 1001 to

23% is equivalent to 23.100

23or

100% is equivalent to 00.1100

100or

6

Exercise 6: Change the decimal to percent

a. .25 b. .085 c. 1.52

Exercise 7: Change the percent to a decimal

a. 2% b. %2

15 c. 0.3%

Example: A class contains 12 female and 8 male students. What percent of the

class is female?

Solution: There are a total of 20 students in the class.

12 out of 20 or 20

12 are female

%60100

60

20

12 of the class is female

Exercise 8: Find the following

a. 30% of 75 b. 4% of 196 c. 120% of 300

Example: Suppose 60% of a class is female. Of these females, 20% are

freshman. What percent of the class is female upper class students?

Solution: If 20% of the females are freshman, then 80% of females are upper

class students. 80% of 60%= 48.60.80.

so 48% of the class are female upper class students

Example: Suppose we know that 40% of a class or 16 students are male. How

many students are in the class?

Solution: We know 40% of (the number) is 16

That is 1640. numberthe

So 4040.

16numberthe students

Example: Before graduating, Joe worked in a convenience store and earned

$5.85/hr. He started a new job that is paying him $18.95/hr. What is the

percent increase in his hourly rate?

Solution: original

increaseactualincrease % so

85.5

85.595.18 or 224% after

rounding.

7

Ratio

The ratio of two numbers a and b is written a to b, a:b or b

a

We can think of this as a fraction in order to simplify it.

The ratio 10:5 is equivalent to 2:1.

The ratio 32.5 to 5 is equivalent to 65 to 10 since 10

65

2

2

5

5.32

5

5.32

► To show two ratios are equal use the following:

D

C

B

A if and only if BCAD

Example: Show that 12.5 : 5 is equivalent to the ratio 5:2

Solution: Since 552525.12 , 2

5

5

5.12 i.e. 12.5: 5 is

equivalent to the ratio 5:2.

Example: How much water should be mixed with 7 cups of salt to make a

12% saline solution?

Solution: We have the proportion 100

127

x

total

part

Multiply to get 70012 x

3

158x

So we need 3

1517

3

158 cups of water.

Exponents

►When working with algebraic expressions involving exponents, the following

properties of exponents are used:

1. baba xxx 4. aaa

yxxy

2. ba

b

a

xx

x 5. a

aa

y

x

y

x

3. abba xx 6. 10 x for any 0x

8

Exercise 9: Simplify each of the following expressions using the rules of

exponents

a. 243x b. 5

10

2

4

ab

a c.

322

25

25

16

xyx

yx

Exercise 10: Find the product 26073 125 zxyzyx

Algebraic Expressions

Most of Algebra involves manipulating expressions that contain variables.

Example: Evaluate the expression using the indicated value of the variable

2513 kforkk

Solution:

52123 substitute -2 in the expression for k

5216

37 = -21

Exercise 11: Evaluate the expression using the indicated value of the variable

a. 2362 2 xforxx b.

414

15

xforx

x

Exercise 12: Combine the terms below and simplify the result.

a. 2834 xx

b. baba 34216

c. 10

13

5

4

xx

Linear Equations in One Variable

A linear equation in one variable can have a unique solution, no solution or

infinitely many solutions. When it has no solution, the equation is called

inconsistent.

Example: Solve the following equations by isolating the variable.

214152 xx

Solution: xxxx 42144152 Add 4x to both sides

21156 x

152115156 x Add 15 to both sides

9

366 x

366

16

6

1

x Multiply both sides by

6

1

6x The equation has a unique solution

Example: Solve the equation 34416 xx

Solution:

121616 xx Distribute the 4

xxxx 1612161616 Add -16x to both sides

120 The equation is inconsistent.

Example: Solve the equation 510625310 xxxx

Solution:

5201215310 xxxx Clear the parentheses by

distributing

15131513 xx Combine like terms

0 = 0

Since the last statement is always true, every real number will be a

solution.

Linear Equations in Two Variables

A solution for an equation in two variables is a replacement for each of the

variables so that when substituted, the resulting statement is true. We generally

write the solution as an ordered pair. For example, 5,2 is a solution of the

equation 113 yx . When the variable x is replaced by 2 in the equation and

the variable y is replaced by 5, the resulting statement is true. The linear

equation in two variables has an infinite number of solutions. The graph of these

solutions is a line in the x-y plane.

Example: The formula in two variables gives the conversion between

Fahrenheit and Celsius temperatures. FytoCx

325

9 xy

Find the temperature in degrees Celsius that corresponds to F212

Solution: 325

9212 x Substitute the value 212 for y

10

x5

9180 Add -32 to both sides

x100 Multiply both sides by 9

5

A solution of the equation is 212,100

The Graph of a Line

We can graph the solutions of any linear equation on a set of axes called the

Euclidean Plane. In the case of the linear equation, the solutions form a line.\We

can measure how much and in which direction the line tips using the slope.

The slope-intercept form of the equation is : bmxy

If the equation is in this form, the line has slope m and y-intercept (0, b)

The y-intercept is the point where the line intersects the y-axis.

Example: In the graph below, find the x and y intercepts and the slope.

Example: Find the slope and y-intercept of the line with equation:

25105 yx

Solution: Write the equation in slope-intercept form: 2

5

2

1 xy

The slope is 2

1 and the y-intercept

2

5,0

Parallel lines have identical slopes. Lines are perpendicular if their slopes have

the relationship

2

1

1

mm

Example: Find the slope of a line that is perpendicular to the line

25105 yx

Solution: The line 25105 yx has slope 2

1 .

Any line perpendicular to it will have slope m= - 2.

Solution:

x intercept: (-12, 0)

y intercept: (0, 6)

Slope is 1/2

11

Algebra I

The Algebra I section of the test measures your ability to perform basic

algebraic operations and to solve problems that involve algebraic concepts.

Only those students attempting to place into Math 108, Math 112, Math 125,

Math 113, CS 112 will be given questions from the Algebra I section. In

addition to all material in the Basic Algebra, the following exercises and

examples provide a refresher of the material that is tested in the Algebra I

section.

Linear Inequalities in One Variable

Unlike linear equations, linear inequalities have an infinite number of solutions.

We can use a number line or interval notation to indicate the solutions. Any real

number can be added to or subtracted from both sides of the inequality. We can

multiply or divide both sides by any nonzero real number but, if we multiply or

divide by a negative number, the inequality sign must be reversed.

Example: Solve the inequality 3 £-9

5x+ 2

Subtract 2 from both sides: 1£-9

5x

Multiply both sides by -5/9: -5

9³ x

The solution is the interval:

]

Absolute Value Equations and Inequalities

The absolute value of a number is defined to be its distance from 0 on the

number line.

0

0

xforx

xforxx

Example: Solve the equation for x 75315 x

Solution: 75315 x or 75315 x

20x or 30x

Example: Solve the inequality 1552 x

Solution: 1552 x or 1552 x

10x or 5x

12

Distance Formula

► The distance between two points 11, yx and 22 , yx is given by

the formula: 212

2

12 yyxxd

The formula uses the Pythagorean Theorem

Exercise 1: Find the distance between the points 5,2 and 8,1

Systems of Linear Equations

We sometimes group equations together and investigate whether they have a

common solution. This is called a system of equations. The system of linear

equations has a solution if there is an ordered pair yx, that satisfies each

equation in the system. A system of linear equations can have exactly one

solution, no solution or infinitely many solutions. There are several methods

available to solve systems of linear equations. The next example illustrates the

Elimination Method.

Example: Solve the system

2489

4823

yx

yx

Solution:

4823 yx multiply by -3: 14469 yx

2489 yx 2489 yx

Add: 16814 y

Solve for y: 12y

The value 12 is substituted for y in either of the original equations.

481223 x Solve the resulting equation for x

48243 x

243 x

8x The solution to the system is (8, 12)

Exercise 2: Solve the systems

a.

1025

53

yx

yx b.

1026

23

yx

yx

13

Linear Inequalities in Two Variables

Linear inequalities in two variables can be solved by graphing. The solution is

an infinite set of ordered pairs in the plane. We can indicate the solution by

shading the region that contains these ordered pairs.

Example: Solve the inequality 5y ³ 3x-15

Solution: Graph the line that results from replacing the inequality sign with an

equal sign. Here, graph the line 5y = 3x-15 This line partitions the plane into

three sets:

The solutions of 5y > 3x-15 5y < 3x-15 5y = 3x-15

In the inequality, test an ordered pair from the plane that does not lie on the line.

Using (0,0), we get: 5(0) ³ 3(0)-15 which is a true statement.

Shade the half-plane of points that satisfy the inequality. See the graph below.

In a similar way we can solve a system of inequalities. The graph below shows

the solutions to the system

5y ³ 3x -15

y £ 0

x ³ 0

ì

íï

îï

Multiplying Algebraic Expressions

We use the Distributive Law to multiply algebraic expressions.

Example: Find the product 1026 xx

Solution: 1026 xx = 10626 xxx

= xx 6012 2

14

To multiply 2 binomial factors, we apply the Distributive Law twice and

combine like terms.

Example: Find the product 127 xx

Solution: 127 xx = 12712 xxx

= 172712 xxxx

= 7142 2 xxx

= 7132 2 xx

The result of this multiplication is called a quadratic expression. We “undo” the

multiplication in a process called factoring.

Factoring Algebraic Expressions

The next two examples show the factoring process. In the first, we factor out the

highest common factor among the terms. In the second, we write the quadratic

expression as a product of 2 binomial factors.

Example: Factor the expression xx 2416 2

Solution:

xx 2416 2 = 2416 xx The terms have the factor

x in common

= 328 xx They also have a factor of

8 in common

Example: Factor the expression 40182 tt

Solution:

40182 tt = 220 tt

Special Factorings:

))((22 bababa

))(( 2233 babababa

))(( 2233 babababa

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator

involve variables. We can reduce, add, subtract, multiply and divide these

rational expressions using the rules of fractions. You may want to review the

rules of fractions in the Basic Algebra section on page 6.

15

Example: Reduce the rational expression 103

7352

xx

x

Solution: In order to reduce the expression we must factor both the numerator

and the denominator, then cancel any factors common to both.

103

7352

xx

x=

52

57

xx

x

=

52

57

xx

x Factor the numerator

= 2

7

x Cancel the common factor

Example: Find the sum 3

1

9

342

x

x

x

x

Solution:

3

1

9

342

x

x

x

x=

33

13

33

34

xx

xx

xx

x Use a common denominator

=

33

3434 2

xx

xxx

= 92

2

x

x

Example: Simplify the expression

ba

x

1

5 2

Solution: Using the rules of dividing fractions we get:

baxba

x

22 51

5

More Exponents

►The definition of exponents is extended to include all rational numbers:

a

a

xx

1

b ab

a

xx

Exercise 3: If x= 3

2 find the value of x-3

Exercise 4: Find the value of 2

1

3

2

927

16

Radicals

When simplifying algebraic expressions involving radicals, the following

properties of radicals are used:

1. nnn xyyx

2. n

n

n

y

x

y

x where 0y

3. nmm n xx

Example: Find the product 5156

Solution: 5156 = 55515616

= 551

Exercise 5: Find the sum: 7328

Exercise 6: Simplify each of the following expressions

a. 568 ba b. 2

37

7

28

uv

vu c. 3 1558 yx

Exercise 7: Rationalize the denominator to simplify: 23

22

Quadratic Equations

An equation of the form 02 cbxax is called a quadratic equation. The

equation will have two, one or no real number solutions. There are several

methods available to find the solutions of a quadratic equation. The following

examples illustrate these methods.

Example: Solve the equation 202 2 x

Solution: 202 2 x

102 x Isolate the term involving the radical

10x Take the square root of both side of the equation

17

Example: Solve the equation 25442 xx

Solution: 25442 xx

02142 xx bring all terms to one side

037 xx factor the left side

07 x or 03 x set each factor equal to zero

7x or 3x the equation has 2 solutions

Example: Solve the equation 0169 2 xx

Solution: In the quadratic formula: a

acbbx

2

42 we use

9a , 6b and 1c

92

19466 2

x =

18

6 since 042 acb

we get only one solution 3

1x to the equation.

►The expression acb 42 is called the discriminant. It tells us how many

solutions a quadratic equation will have.

If 042 acb the equation has 2 real number solution

If 042 acb the equation has 1 real number solution

If 042 acb the equation has no real number solution

Exercise 8: Solve the equations

a. 0494 2 x

b. 0592 2 xx

c. 0642 xx

Function Notation

Functions can be defined using a list of ordered pairs, a graph, a table, or an

equation.

The following equation defines a function: 12 xxf .

Each value of x is paired with the value xf .

8133 2 f so the pair 8,3 is in the function f.

9911010 2 f so the pair 99,10 is in the function f.

99110102

f so the pair 99,10 is in the function f.

18

Example: For the function 3

12

x

xxf find 2f and 1bf

Solution:

31

3

32

1)2(2

2

f

4

2

3)1(

1)1(1

22

b

bb

b

bbf

Complex Rational Expression

A complex rational expression is one in which a rational expression appears in

the numerator and/or denominator. Simplifying the complex rational expression

is equivalent to finding the quotient of the expression in the numerator with the

expression in the denominator.

Example: Simplify the complex rational expression

yx

xy

11

1

Solution: Combine the terms in the denominator first.

xy

xy

yx

11

Next, find the quotient of the numerator with the denominator

xy

xy

xy

1=

xy

xy

xy

1

= xy

1

Equations with Rational Expressions

Given an equation involving rational expressions we can find an equivalent

equation with no rational expressions. To find the equivalent equation we

multiply the entire equation by the least common denominator (LCD). After

solving the equivalent equation, we must check the solution in the original to see

if it makes any denominator equal to zero. If so, we discard that solution.

Example: Solve the equation xx 2

3

5

11

Solution: The LCD of the rational expressions is 10x

x

xxx

x2

310

5

110

110 Multiply equation by 10x

19

15210 x Solve the resulting equation

52 x

2

5x

Example: Solve the equation xx

3

1

3

4

Solution: The LCD of the rational expressions is 3x

xxxx

x 33

13

3

43 Multiply equation by 3x

234 xx Solve the resulting equation

043 2 xx Use the quadratic formula

32

)4(3411 2

x

32

491

x

3

41 andx

Exercise 9: Solve the equation 32

56

1

32

3

22

xx

x

x

x

x

Algebra II

The Algebra II section of the test measures your ability to solve problems that

involve more advanced Algebra than the previous section. Only those students

attempting to place into Math 113 and CS 112 will be given questions from the

Algebra II section. The following exercises and examples provide a refresher of

material that is tested in the Algebra II section.

More on Functions

A function is a set of ordered pairs where each first component is paired with

exactly one second component. The set of first components is called the domain

and the set of second components is called the range.

Unless otherwise stated, the domain of the function 12 xxf , is all real

numbers. Any real number a can be paired with another real number using the

equation 12 xxf . The pair can be written 1, 2 aa .

20

Suppose we have the function t

tg3

. We say tg is undefined at 0t

since the expression 0

3 is undefined. On the other hand, tg is defined at any

real number other than 0. The domain of tg is all real numbers except 0t .

Suppose we have the function 5 xxf . In order for the expression

5x to represent a real number, the expression 5x must not be negative.

For this function the domain is all real numbers 5x .

►The average rate of change for a function f(x) is the change in f(x) over the

change in x.

x

xfchangeofrateaverage

)(

Alternatively, it is the slope of the secant line passing through the two points

(x1, f(x1)) and (x2, f(x2))

Exercise 1: Let 62

1

xxf ; Find each of the following:

a. the domain of xf b. the range of f(x)

c. the average rate of change from x= -1 to x=5

Exercise 2: Let x

xg

3

1 ; Find each of the following:

a. the domain of xg b. the range of g(x)

c. the average rate of change from x= -2 to x=2

►We can compose two functions by applying one right after the other.

Example: For the functions f and g defined in the exercises 1 and 2, find

a. 6gf b. 2fg

Solution: 6gf =

)6(3

1f =

3

1f =

20

3

2fg =

6)2(2

1g =

2

1g =

2/5

1 or 5

2

Example: The function values of f and g are given in the table below. Use them

to find a. 0gf b. 3fg

21

x f(x) g(x)

0 -3 3

1 0 2

2 -2 1

3 1 0

Solution: a. 0gf = 3f =1

b. 3fg = 1g =2

►Some functions have an inverse. The inverse of a function will interchange

the real numbers in the ordered pair. If ba, is in the function f then ab, is in

1f , the inverse of f . Keep in mind, that not all functions have inverses.

The functions 53 xxf and 3

51 x

xf are inverses.

Notice f contains the pair (3,14) and f-1

contains the pair (14, 3). This is true

for all ordered pairs of the function f.

Graphs of Functions

We can graph the set of ordered pairs of a function on a set of axes just as we

do with linear equations in 2 variables.

The graph of a quadratic function cbxaxxf 2)( is a parabola.

The vertex of the parabola is located at

)

2(,

2 a

bf

a

b

This point on the graph is either a minimum or a maximum for the function. The

leading coefficient a, determines which. For a<0, the function takes on a

maximum at a

bx

2

and the range of the function is

a

bfyy

2.

For a>0, the function takes on a minimum at a

bx

2

and the range of the

function will be

a

bfyy

2.

Example: In the graph of 74)( 2 xxxf below, find the x and y

intercepts and the vertex.

22

► To graph the function baxfcy )( we shift the graph of f(x)

right a units (for a>0), left |a| units for a<0)

up b units for b>0, down |b| units for b<0

vertically stretch the graph of f(x) by a factor of c, c>1

vertically shrink the graph of f(x) by a factor of c, for 0<c<1

Example: The graph above can be viewed as a shift of 2xy

(2 units right and 3 units up). The function can be written in the form:

3)2()( 2 xxf

Transcendentals

The test measures your ability to work with exponential, logarithmic and

trigonometric functions of the right triangle and the unit circle. The graphs of

these functions and some common trigonometric formulas are given in the next

few pages.

Exponential Functions

An exponential function is one in which the variable occurs in the exponent.

The base determines whether the function is increasing or decreasing.

1)( aforaxf x 1)( aforaxf x

Domain: all real numbers Domain: all real numbers

Range: y>0 Range: y>0

Solution:

y intercept: (0, 7)

x intercept: none

vertex: (2, 3)

note: (2, 3) is the minimum of f(x)

23

Logarithmic Functions

A logarithmic function is the inverse of an exponential function.

Example: Evaluate: a. 8log2 b. 1log5

c. 001.log10

Solution: a. 38log2 since 823 b. 01log5 since 150

c. 3001.log10 since 001.10 3

Example: Solve the equation for x: 103 xe

Solution: Isolate x by taking the log of both sides. Here use ln the natural log.

10lnln 3 xe

10ln3 x

10ln3x 6974.

Example: Solve the equation for x: 6)13(log 4 x

Solution: Use the inverse function to isolate x 6)13(log

44 4 x

6413 x

4083x

Angle Measure

An angle can be measured in degrees or in radians. A full revolution is 360 or

2 radians. We use this information to convert from one measure to the other.

The following proportion shows the relationship between x degrees and y

radians. 2

360

y

x

Example: Find the degree measure of an angle whose radian measure is 3

2

Solution: 2

360

3/2

x

3/23602 x so 120x

This is the natural log

function, (the inverse of

y=ex)

f(x)= ln x

Domain: x>0

Range: all real numbers

24

Right Triangle Definitions

hyp

oppsin

opp

hypcsc

hyp

adjcos

adj

hypsec

adj

opptan

opp

adjcot

Definitions of Circular Functions

Special Angles

Degree Radian sin θ cos θ tan θ csc θ sec θ cot θ

0 0 0 1 0 -- 1 --

30 π/6 2/1 2/3 3/3

2 3/32

3

45 π/4 2/2

2/2

1 2 2 1

60 π/3 2/3

2/1 3 3/32

2 3/3

90 π/2 1 0 -- 1 -- 0

Sum and Difference Formulas

vuvuvu sincoscossinsin

vuvuvu sinsincoscoscos

25

Graphs of the Trigonometric Functions

f(x)=sin(x) period:

f(x)=cos(x) period:

f(x)=tan(x) period:

Exercise 1: Find the period of

a. x2sin b.

4cos

x c.

3tan

x

Exercise 2: Which has the shortest period?

a. xy cos b. xy cos2

1 c. )2cos( xy d. xy

2

1cos

26

Inverse Trigonometric Functions

The inverse sine function, written )(sin 1 x or )arcsin(x , has

domain [-1,1] and range

2,

2

. It is defined to be the number in the

interval

2,

2

whose sin is x.

In a similar way, )(cos 1 xor )arccos(x is defined to be the number

in the interval ,0 whose cosine is x. The inverse tangent function, written

)(tan 1 x or )arctan(x , is defined to be the number is the interval

2,

2

whose tangent is x.

Exercise 3: Find each of the following:

a.

3

3arctan b. )

3arccos(cos

c. )0arcsin(

27

Double Angle Formulas

uuu cossin22sin

1cos2sin21sincos2cos 2222 uuuuu

Law of Sines

c

C

b

B

a

A sinsinsin

Law of Cosines

Cabbac cos2222

Pythagorean Identity

1cossin 22 xx

Exercise 4: In the triangle below, find the length of the side marked with x.

Exercise 5: In the triangle below, find the length of the side marked with x.

Exercise 6: If 13

5sin A and

12

5tan A , find Acos

28

Answers to Exercises

Basic Algebra

1. 56 2. 167 3. $63.00

4. (a) (b) 16

1 (c)

20

173 (d)

3

210

5. (a) 8

17 (b)

20

91 (c)

24

11 (d)

3

21

6. (a) 25 % (b) %2

18 (c) 152 %

7. (a) .02 (b) .055 (c) .003 8. (a) 22.5 (b) 7.84 (c) 360

9. (a) 89x (b)

5

92

b

a (c)

45

2

y

10. 213460 zyx

11. (a) 23 (b) 21/80

12. (a) 144 x (b) ba 1020 (c) 10

15 x

Algebra I

1. 18 or 23

2. (a) (0, -5) (b) inconsistent system

3. 8

27

4. 33 or 27 5. 75

6. (a) bba 22 23 (b) vu32 (c)

3 252 xxy

7. 624

8. (a) 2

7,

2

7 (b)

2

1,5 (c) 102,102

9. 6,2

1

Algebra II

1. (a) all real numbers except 3x (b) all real numbers except y=0

(c) -1/32

2. (a) all real numbers (b) all real numbers y>0

(c)

√

√

3

21

3x

29

Transcendentals

1. (a) (b) (c) 3

2. c

3. (a) 30 (b)

3

(c) 0

4. 70sin2

1 5. 12 6.

13

12

8