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Math 098 – Chapter 1

Section 1.1 – Basic Concepts about Triangles

1) Conventions in notation for triangles - Vertices with uppercase - Opposite sides with corresponding lower case

2) Pythagorean theorem

In a right triangle, the sum of the squares of the _________ equals the square of the ___________________

3) Given 2 sides of a right triangle, find the other one. a) Given the two legs

Find the hypotenuse if the legs are 10cm and 8 cm. Sketch, label and show work.

b) Given the hypotenuse and one of the legs. If the hypotenuse is 40 in. and one of the legs is 30 in., find the other leg. Sketch, label and show work.

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Section 1.1 – Basic Concepts about Triangles

4) Sum of the three angles in a triangle A + B + C = …………..

What can you say about the sum of the two acute angles in a right triangle?

5) Given 2 angles of a triangle, find the third one a) A = 47˚ and B = 120˚

b) In a right triangle, one of the acute angles is 56˚, find the other one.

HWK – Read the examples in the book and COMPLETE PROBLEMS 5-14 ON PAGE 4 of your book

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Section 1.2 – Right Triangle Trigonometry

1) Definitions of the six trigonometric functions

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Section 1.2 – Right Triangle Trigonometry

1-a - Find the exact values of all the trigonometric functions of A and B. x = 36 y = 60 z = 48

1-a - Find the exact values of all the trigonometric functions of A and B. x = 82 y = 80 z = 18

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Section 1.2 – Right Triangle Trigonometry

2) Use the calculator in DEGREE mode to find trig functions of some acute angles a) Sin(50˚) (b) Cos(75˚) (c) Tan(65˚)

3) Solve right triangles given one side and one acute angle a. One acute angle is 38˚and the hypotenuse is 10 cm; find all other sides and angles

b. One acute angle is 79˚and one of the legs is 8 in.; find all other sides and angles

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Section 1.2 – Right Triangle Trigonometry - continued

4) Given one trigonometric function of an acute angle, find the other trig function values

a. Given that sin(A) = 2/3, find all other trig function-values of A.

b. If tan(B) = 8/5, find all other trig-function-values of B.

c. If cos(A) = 5/7, find all other trig-function-values of A.

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Section 1.2 – Right Triangle Trigonometry - continued

5) The inverse trigonometric functions

6) Use the calculator in DEGREE mode to find angles when the trig function of the angle is given

(a) Find angle A if Sin(A) = 0.876 (b) find angle B if cos(B) = 0.485

(c) Find angle β if tan(β) = 4.56 (d) What is angle E if csc(E) = 1.45?

(e) Find angle F if cot(F) = 7.9 (f) What is angle α if sec(α) = 3.25?

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Section 1.2 – Right Triangle Trigonometry - continued

7) Solve right triangles given two sides of the triangle

a) The two legs of a right triangle are 5cm and 9 cm. Find the hypotenuse and the two acute angles.

b) Solve a right triangle if the hypotenuse is 15 in. and one of the legs is 10 in.

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Section 1.2 – Right Triangle Trigonometry - continued

8) Special triangles

9) Evaluate the six trigonometric functions of “special” acute angles

10) HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGES 10, 11, and 12

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Section 1.3 – Trig Functions of any Angle

1) Angles in standard position

- Vertex at the origin - Initial ray (side) coincides with the x-axis - Terminal ray (side)

2) Positive and negative direction of angles

- Positive direction: counterclockwise - Negative direction: clockwise

3) Recalling basics of the coordinate system

4) Quadrantal or boundary angles

Both sides of the angle lie on the quadrants

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Section 1.3 – Trig Functions of any Angle

5) Reference angle - Is the acute angle formed by the terminal ray and the x-axis

Give the reference angle for each of the following angles:

(1) 340◦ (2) 128◦ (3) 250◦ (4) 280◦

6) Coterminal angles - They have the same terminal ray

Give two positive and a negative coterminal angle for each of the following:

(1) 30◦ (2) 130◦ (3) 270◦

7) Sigs of trigonometric functions of angles in the four quadrants

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Section 1.3 – Trig Functions of any Angle - continued

8) Given a point on the terminal side of the angle, find all trig functions of the angle (1) P(-3, 5)

(2) P(5, -2)

(3) P(-2, -3)

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Section 1.3 – Trig Functions of any Angle - continued

9) Given the trig function of an angle in a specific quadrant, find all other trig functions

a) Given that sinα = 2/3 and 90˚ < α < 180˚, find all other trig functions of the angle α

b) Given that cosα = -3/4 and 180˚ < α < 270˚, find all other trig functions of the angle α

c) Given that tanα = - 5/3 and 270˚ < α < 360˚, find all other trig functions of the angle α

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Section 1.3 – Trig Functions of any Angle – continued

10) Find trig function-values of special angles in all four quadrants

Sketch the angle in the correct quadrant, project towards the x-axis, put + and – on the three sides of the obtained triangle, find the reference angle. Give the trig-function-value (based on the RA) and the sign (based on the definition and signs of sides)

a) Find sin 120˚ (b) Find cos 225˚

b) Find tan 240˚ (b) Find cot 315˚

c) Find sec 210˚ (b) Find csc 150˚

11) HWK – read all the book-examples and COMPLETE ALL PROBLEMS ON PAGE 21

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Section 2.1 – Trigonometric Identities – Take Home Assignment

1) Do the following: a) Go to my website: http://faculty.montgomerycollege.edu/maronne/ b) Go to math 098 c) Click on Professor Yagodich’s videos (2rd row – yellow row) d) Scroll down to the HOMEWORK 6 section and click on the video – Basic Trig Identities e) Write them on this page:

• The reciprocal identities

• The quotient (or ratio) identities

• The Pythagorean identities

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Section 2.1 – Trigonometric Identities – Take Home Assignment - continued

2) Now, on the same Homework 6 section, watch the following video Algebraic manipulation part I – (~ 9 min)

Write here the four examples involving trigonometric functions that are given on this video. Show all steps to simplify each problem. Be neat! You will need to refer to these examples when you do the back next page.

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Section 2.1 – Trigonometric Identities – Take Home Assignment - continued

3) Now use techniques similar to the ones shown on the video to show the solutions to each of the following 4 problems. SHOW ALL WORK – NEATNESS, PLEASE!

i. Multiply and simplify: sinx (cotx – cscx) ii. Multiply: (cotx + tanx)(cotx-tanx)

iii. Factor: 2 2 4sin cos sinx x x+

iv. Factor: cos cos sincos cos cot

x x xx x x−−

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Section 2.1 – Trigonometric Identities

4) Prove basic identities – show that the left hand side equals the right hand side a) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (b) 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐 𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 = 𝑐𝑐𝑡𝑡𝑠𝑠𝑐𝑐

5) Prove basic identities – show that the left hand side equals the right hand side

a) 𝑐𝑐𝑡𝑡𝑠𝑠𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡 − cos2 𝑡𝑡 = sin2 𝑡𝑡 b) 𝑐𝑐𝑠𝑠𝑠𝑠𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡 − sin2 𝑡𝑡 = 𝑐𝑐𝑐𝑐𝑐𝑐2 𝑡𝑡 c) 𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐 𝑐𝑐𝑡𝑡𝑠𝑠𝑐𝑐 d) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

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Section 2.1 – Trigonometric Identities

6) Prove basic identities – show that the left hand side equals the right hand side

a) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 b) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑡𝑡𝑠𝑠2𝑐𝑐 = 𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 c) 𝑐𝑐𝑡𝑡𝑠𝑠𝑐𝑐 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 d) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐 𝑐𝑐𝑡𝑡𝑠𝑠𝑐𝑐 = 𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 e) (𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 − 1)(𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐 + 1) = tan2 𝑐𝑐

7) HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGE 25 8) HWK – COMPLETE THE MIDCOURSE REVIEW, PAGE 26 9) HWK – COMPLETE THE PRACTICE TEST, PAGE 27

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Chapter 3.1 – Radian Measure

1) Definition of radian

Θ = 1 radian when the arc equals to the radius

2) Relationship between degrees and radians Working on the unit circle, when radius = 1, the radian measure is simply the length of the arc associated with the central angle θ. The length of a full unit circle is 2π (recall circumference formula: C = 2πr = 2π(1) = 2π)

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Section 3.1 – Radian Measure

3) Convert from degrees to radians Give exact answers in terms of pi, then an approximation to two decimal places. Sketch.

a) 120˚ (d) 135˚

b) 210˚ (e) 300˚

c) 170˚ (f) 280˚

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Section 3.1 – Radian Measure

4) Convert from radians to degrees, show sketch a) 11pi/6 (e) 1 rad.

b) 5pi/4 (f) 5 rad

c) 4pi/3 (g) 4 rad.

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Section 3.1 – Radian Measure

5) Trig-function-values of angles given in radians a) Sin (7pi/6) (e) Sec(5pi/6)

b) Cos(2pi/3) (f) Csc(7pi/4)

c) Tan(3pi/4) (g) sin(-pi/4)

d) Cot(5pi/3) (h) cos(-4pi/3)

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Section 3.1 – Radian Measure

6) Trig–function-values of boundary (quadrantal) angles

(a) Sin(6pi) (d) Cos(-3pi)

(b) Tan(-pi/2) (e) Cot(pi)

(c) Sec(3pi/2) (f) Csc(9pi/2)

HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGES 32 and 33

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Chapter 4 – Graphs of Trigonometric Functions

Section 4.1 – The Unit Circle

1) What is the unit circle? 2) What do the coordinates of a point on the unit circle represent?

3) Find the coordinates of the point of intersection of theta with the unit circle.

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Section 4.1 – The Unit Circle continued

4) Find the coordinates of the point of intersection of theta with the unit circle.

5) HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGE 36

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Section 4.2 – Graphs of Trigonometric Functions – Graphs of y = A sin(x)

1) Graph y = sin(x) a. Label all tic-marks. b. Graph from [0, 2pi)

c. Give the domain, range, amplitude and period

2) Give the amplitude and graph 𝑓𝑓(𝑐𝑐) = 2 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐 from [0,2pi] using the 5-key-points. Then, expand through 4pi. Label quadrantal angles along the x-axis and the maximum and minimum y-values

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Section 4.2 – Graphs of Trigonometric Functions – Graphs of y = A cos(x)

3) Graph y = cos(x) a. Label all tic-marks. b. Graph from [0, 2pi)

c. Give the domain, range, amplitude and period

4) Give the amplitude and graph 𝑓𝑓(𝑐𝑐) = 23𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 from [0,2pi] using the 5-key-points. Then, expand to the left

through (- 4pi). Label quadrantal angles along the x-axis and the maximum and minimum y-values

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Section 4.2 – Graphs of Trigonometric Functions

5) Graph of y = tan(x) a. Label all tic-marks. b. Graph from [0, 2pi]

c. Give the domain, range, amplitude and period

d. Graph y = tan(x) over the interval [-2pi, 2pi]; label quadrantal angles.

(1) HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGE 39

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Section 5.1 - Inverse Trigonometric Functions

1) Inverse sine, inverse cosine and inverse tangent functions a. Domain and range

2) Simple evaluations 3) HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGE 44

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Section 5.2 – Trigonometric Equations

Solve basic trig equations over the interval [0, 2pi)

1) Find the angle 𝜶𝜶 over the interval [0, 2pi) if 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 = 𝟏𝟏𝟐𝟐

a) Procedure: In what quadrants is the sine positive? Sketch the two angles in the corresponding quadrants. Based on the given value of ½, what is the reference angle? Label it on the graph. What are the two solutions? In degrees? In radians?

b) Use the graphical approach – Graph = sin x and y = 1/2

c) Solve the equation over the whole domain

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Section 5.3 - Solve basic trig equations over the interval [0, 2pi)

2) Find the angle 𝜶𝜶 over the interval [0, 2pi) if 𝐜𝐜𝐜𝐜𝐬𝐬𝐬𝐬 = √𝟐𝟐𝟐𝟐

a) Procedure: In what quadrants is the cosine positive? Sketch the two angles in the corresponding

quadrants. Based on the given value of √𝟐𝟐𝟐𝟐

, what is the reference angle? Label it on the graph. What are the two solutions? In degrees? In radians?

b) Use the graphical approach – Graph y = cos x and y = √𝟐𝟐𝟐𝟐

~ 𝟎𝟎.𝟕𝟕

c) Solve the equation over the whole domain

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Section 5.3 - Solve basic trig equations over the interval [0, 2pi)

3) Find the angle 𝜶𝜶 over the interval [0, 2pi) if 𝐭𝐭𝐭𝐭𝐬𝐬𝐬𝐬 = √𝟑𝟑 a) Procedure: In what quadrants is the tangent positive? Sketch the two angles in the corresponding quadrants.

Based on the given value of √𝟑𝟑, what is the reference angle? Label it on the graph. What are the two solutions? In degrees? In radians?

b) Use the graphical approach – Graph y = tan x and y = √𝟑𝟑 ~ 𝟏𝟏.𝟕𝟕

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Section 5.3 - Solve basic trig equations over the interval [0, 2pi)

4) Solve the following equations over the interval [0, 2pi) using the procedure outlined on the previous pages. Then, show the graphical approach.

a. 𝒄𝒄𝐜𝐜𝐬𝐬𝐬𝐬 = −√𝟑𝟑𝟐𝟐

b. 𝒔𝒔𝒔𝒔𝒔𝒔𝐬𝐬 = −√𝟑𝟑𝟐𝟐

c. 𝒕𝒕𝒕𝒕𝒔𝒔𝐬𝐬 = −√𝟑𝟑𝟑𝟑

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Section 5.3 - Solve basic trig equations over the interval [0, 2pi)

5) Solve the following equations over the interval [0, 2pi); then show the graphical approach. a. 𝒄𝒄𝐜𝐜𝐬𝐬𝐬𝐬 = −𝟏𝟏 b. 𝒔𝒔𝒔𝒔𝒔𝒔𝐬𝐬 = −𝟏𝟏 c. 𝒕𝒕𝒕𝒕𝒔𝒔𝐬𝐬 = 𝟎𝟎 d. 𝒄𝒄𝐜𝐜𝐬𝐬𝐬𝐬 = 𝟎𝟎 e. 𝒔𝒔𝒔𝒔𝒔𝒔𝐬𝐬 = 𝟎𝟎 f. 𝒄𝒄𝐜𝐜𝐭𝐭𝐬𝐬 = 𝟎𝟎 g. Secα = 1 h. Cscα = 1

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Section 5.2 – Trigonometric Equations - Using the calculator to find the reference angle.

6) Solve the following equations over the interval [0, 2pi) using the procedure outlined on the previous pages. In this case, we are not familiar with the given trig-function-value and we need to use the calculator to find the reference angle. Then, show the graphical approach.

a. 𝒕𝒕𝒕𝒕𝒔𝒔𝐬𝐬 = 𝟐𝟐.𝟓𝟓𝟓𝟓

b. 𝒄𝒄𝒄𝒄𝒔𝒔𝐬𝐬 = 𝟎𝟎.𝟕𝟕𝟓𝟓

c. 𝒔𝒔𝒔𝒔𝒔𝒔𝐬𝐬 = 𝟎𝟎.𝟒𝟒

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Section 5.2 – Trigonometric Equations - Using the calculator to find the reference angle.

7) Solve the following equations over the interval [0, 2pi). Isolate the trig function first, then use the procedure outlined on the previous pages.

a. 𝟒𝟒 𝒔𝒔𝒔𝒔𝒔𝒔𝐬𝐬+ 𝟑𝟑.𝟖𝟖𝟓𝟓 = 𝟎𝟎

b. 𝟑𝟑 𝒔𝒔𝒔𝒔𝒔𝒔𝐬𝐬 − 𝟒𝟒.𝟓𝟓 = 𝟎𝟎

c. 𝟓𝟓𝒄𝒄𝒄𝒄𝒔𝒔𝐬𝐬 + 𝟑𝟑.𝟑𝟑 = 𝟎𝟎

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Section 5.2 – Trigonometric Equations - Using the calculator to find the reference angle.

8) Solve the following equations over the interval [0, 2pi). Isolate the trig function first, then use the procedure outlined on the previous pages.

a. 𝟓𝟓𝒄𝒄𝒄𝒄𝒔𝒔𝐬𝐬 + 𝟏𝟏𝟎𝟎 = 𝟎𝟎

b. 𝟓𝟓𝒕𝒕𝒕𝒕𝒔𝒔𝐬𝐬+ 𝟏𝟏𝟓𝟓 = 𝟏𝟏

c. 𝟑𝟑𝒄𝒄𝒄𝒄𝒔𝒔𝐬𝐬 + 𝟐𝟐.𝟓𝟓𝟐𝟐 = 𝟎𝟎

9) HWK – Read the examples in the book and COMPLETE ALL PROBLEMS ON PAGE 46 10) HWK – COMPLETE ALL PROBLEMS ON THE FINAL REVIEW and PRACTICE TEST, PAGES 47 and 48

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