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Aratari, Trigonometry: Chapter 2-Form A Name: 41

Take the Quiz!2.1

1. Graph one cycle of the pure form y = sin x and label the critical points.

–1

1

x

y

2��–2� –�

2. Graph one cycle of the pure form y = cos x and label the critical points.

–1

1

x

y

2��–2� –�

3. Find an equation of the form y = A cos x or y = A sin x that represents the graph.

a. b.

x

y

�–�

–8

–1

1

–2� 2�–2

4

x

y

2�–2�

Aratari, Trigonometry: Chapter 2-Form B Name: 43

Take the Quiz2.1

1. Graph one cycle of the pure form y = cos x and label the critical points.

–1

1

x

y

2��–2� –�

2. Graph y = −3 sin x on −2π ≤ x ≤ 2π . State the range and the x-intercepts.

range

x-intercepts

–3

4

1

x

y

2��–2� –�

3. Find an equation of the form y = A cos x or y = A sin x that represents the graph.

a. b.

–4

5

1

x

y

2�–2�

–3

2

x

y

2�–2�


Take the Quiz2.1–2.2

1. Graph at least one cycle of each of the following and label the critical points for one cycle. Fill in the blanks, ifappropriate, with the amplitude, period, x-intercepts and/or range.

a. y = −4 cos xAmplitude

Period

x-intercepts

–3

4

1

x

y

2��–2� –�

b. y = sin1

4x

Amplitude

Period

Range

x

y

46 Aratari, Trigonometry: Quiz 2.1–2.2 Form A

2. Find an equation in the form y = A sin[B(x − C)], or y = A cos[B(x − C)] that represents the graph. More thanone answer is possible.

a.

–2

2

x

y

2��–2�

b.

–4� 4�

–6

6

3

x

y

–2� 2�



1. Graph one cycle of each of the following and label the critical points. Fill in the blanks with the amplitude, period,x-intercepts and range.

a. y = 2 sin x

Amplitude

Period

Range

x-intercepts

–3

4

1

x

y

2��–2� –�

b. y = cos

(1

2x

)Amplitude

Period

Range

x-interceptsx

y

48 Aratari, Trigonometry: Quiz 2.1–2.2 Form B

2. Find an equation in the form y = A sin[B(x − C)], or y = A cos[B(x − C)] that represents the graph. More thanone answer is possible.

a.

–5

–1

6

x

y

2�

2�–

b.

–3

1

x

y

2��–2� –�



1. Graph a minimum of one period of each of the following. Label critical points and fill in the requested information.

a. y = 5 sin(2x)Period

Amplitude

x-intercepts

x

y

b. y = 3 cos(x − π

2

)Period

Amplitude

Range

Phase Shiftx

y


2. Graph one period of y = tan

(1

4x

).

Period

Asymptotes

x

y

3. Find an equation that represents the circular function graph. (The guide function has been dotted in.)

�–� x

y

2

–12�

2�–



1. Graph at least one cycle of each of the following and label the critical points for one cycle. Where necessary, fill inthe blanks with the requested information.

a. y = 2 sin

(1

4x

)Period

Amplitude

x-intercepts

x

y

b. y = 4 cos(x + π

2

)Period

Amplitude

Phase Shift

x

y


2. Graph one period of y = cot

(1

2x

).

Period

Asymptotes

x

y

3. Find an equation that represents the circular function graph. (The dotted graph represents the guide function.)

�–� x

y

1

–1

–5

4�


Take the Quiz2.5

1. Rewrite each expression to eliminate the inverse notation. Then find the exact value for y in the restricted rangewithout using a calculator.

a. y = arccos

√2

2⇒ cos =

y =

b. y = tan−1

√3

3⇒ tan =

y =

c. y = sin−1

(−

√3

2

)⇒ sin =

y =

d. y = arcsec(−2) ⇒ sec =

y =

2. Use a calculator to find an approximation to four decimal places for each expression.

a. arcsin(−0.87) b. csc−1 5



1. Sketch the graph of each function for two cycles. Label x-intercepts and asymptotes (if applicable).

a. y = tan[2(x + π

4

)]

x

y

b. y = − csc1

4x

x

y

2. Find an equation of a sinusoidal function that represents the graph .

–4

4

x

y

�–2� 2�–�


3. Rewrite each expression to eliminate the inverse notation. Then find the exact value for y in the restricted rangewithout using a calculator. If no value for y is possible, explain why.

a. y = arccos

√3

2b. y = tan−1(−1)

c. y = sin−1

(−

√2

2

)d. y = arccsc

1

2

4. Use a calculator to find an approximation to four decimal places for each expression.

a. arcsin(0.5559) b. cot−1(

5

17

)

Aratari, Trigonometry: Chapter 2 Test-Form A Name: 57

Chapter 2 Test

1–5 Without using a graphing utility, graph two cycles of the given function. Provide the requested information.

1. y = 4 cos(−2x)Period

Amplitude

Range

x

y

2. y = sin

[1

2(x + π)

]Period

Amplitude

Phase Shift

x-interceptsx

y

58 Aratari, Trigonometry: Test Form A

3. y = − cot(4x)Period

x-intercepts

asymptotes

x

y

4. y = 4 csc(x − π

2

)Period

Range

x

y

5. y = cos(π

2x)

+ 2

Period

Range

x

y


6. Find an equation for each circular functions graph.

a. y = b. y =

–3

2

x

y

2�

2�–

–3

x

y

4�–4�

c. y = d. y =

x

y

1

2

–1

–2

2�

23�

2�– –3

3

x

y

2�

2�–

60 Aratari, Trigonometry: Test Form A

7. Without the use of a calculator, find the exact value for y.

a. y = arccos 0 b. y = arctan(−1)

c. y = sin−1(

1

2

)d. y = arcsec

2√3

e. y = cot−1

(√3

3

)f. y = csc−1

(−√

2)

g. y = cos−1(

−1

2

)h. y = sin−1(−1)

i. y = sin

(arctan

(−

√3

3

))j. y = tan

(arccos

15

17

)

8–11 Use your calculator to find an approximation to four decimal places for each expression.

8. sin−1(−0.45) 9. arctan 53.2

10. csc−1(

−13

7

)11. sec(arctan 2.5)

12. Solve the equation 7 cos 4x = 2 for x. Assume the arc restrictions are those specified in the definition for thearccosine.


13. Identify the inverse circular function graph: y =

1–1x

y

2�

2�–

14. The graph below models the body temperature in degrees Fahrenheit for a five day illness.

1 2 3 4 5

98°

100°

104°

Day

a. On which days (to the nearest half-day) will the temperature be 101◦?

b. Approximate the number of days (to the nearest half-day) between the 101◦ temperatures.

c. Approximate the difference in the temperatures (to the nearest degree) between the highest and lowest

temperatures.


Chapter 2 Test


1. y = 3 sin

(−1

4x

)Period

Amplitude

Range

x

y

2. y = cos[2(x + π

4

)]Period

Amplitude

Phase Shift

x-interceptsx

y

64 Aratari, Trigonometry: Chapter 2 Test Form B

3. y = − tan

(1

3x

)Period

x-intercepts

asymptotes

x

y

4. y = 4 sec xPeriod

Range

x

y

5. y = sin(π

3x)

− 1

Period

Range

x

y


6. Find an equation for each of the following circular functions.

a. y = b. y =

1

2

x

y

3�

3�–

4

2

x

y

� 2�2�

6�

c. y = d. y =

�–2� 2� x

y

2

1

–2

�–� x

y

2

–1



a. y = arcsin 1 b. y = arccot√

3

c. y = cos−1(

1

2

)d. y = arccsc

2√3

e. y = tan−1(−1) f. y = sec−1(−√

2)

g. y = sin−1(

−1

2

)h. y = cos−1(−1)

i. y = cos

(arctan

(−

√3

3

))j. y = sin

(arccos

15

17

)


8. cos−1(−0.45) 9. arctan 89

10. sec−1(

−13

7

)11. cot(arctan 2.5)

12. Solve the equation 7 sin 4x = 5 for x. Assume the arc restrictions are those specified in the definition for thearcsine.



1–1x

y

�

14. The graph below models the body temperature in degrees Fahrenheit for a five day illness.

1 2 3 4 5

98°

100°

104°

Day

a. On which days (to the nearest day) will the temperature be 102◦?

b. Approximate the number of days (to the nearest day) between the 102◦ temperatures.

c. Approximate the difference in the temperatures (to the nearest degree) between the highest and lowest

temperatures.

Aratari, Trigonometry: Chapter 2-Form C Name: 69

Chapter 2 Test


1. y = 3 cos

(−1

4x

)Period

Amplitude

Range

x

y

2. y = sin[2(x + π

4

)]Period

Amplitude

Phase Shift

x-interceptsx

y

70 Aratari, Trigonometry: Chapter 2 Test Form C

3. y = − cot

(1

3x

)Period

x-intercepts

asymptotes

x

y

4. y = 4 csc xPeriod

Range

x

y

5. y = cos(π

3x)

− 1

Period

Range

x

y



a. y = b. y =

4

x

y

3�–3�x

y

1

–1

4�

43�

4�–

c. y = d. y =

2

1

x

y

611�

6�–

4

x

y

2��–1



a. y = arccos1

2b. y = arctan 0

c. y = sin−1

(√2

2

)d. y = arccsc 2

e. y = arcsec(−1) f. y = csc−1(−√

2)

g. y = cot−1(

1√3

)h. y = sin−1(−1)

i. y = cos(arctan(−1)) j. y = sin

(arccos

8

17

)


8. sin−1 0.54 9. arccsc 89

10. cos

[tan−1

(−13

7

)]11. tan(arcsec 2.5)

12. Solve the equation 3 cos 5x = 2 for x. Assume the arc restrictions are those specified in the definition for thearccosine.



x

y

2�

2�–

14. The graph below models the tides in feet, x hours after midnight for one day.

1 6 12 18 24

1

2

3

Hours

a. Approximate the times (to the nearest half-hour) for high tide.

b. Approximate the times (to the nearest half-hour) for low tide.

c. Approximate the difference in the water level (to the nearest foot) between high tide and low tide.

Aratari, Trigonometry: Chapter 2-Form D Name: 75

Chapter 2 Test


1. y = 4 sin(−2x)Period

Amplitude

Range

x

y

2. y = cos

[1

2(x − π)

]Period

Amplitude

Phase Shift

x-interceptsx

y

76 Aratari, Trigonometry: Chapter 2 Test Form D

3. y = − tan(4x)Period

x-intercepts

asymptotes

x

y

4. y = 4 sec(x + π

2

)Period

Range

x

y

5. y = sin(πx) + 1Period

Range

x

y



a. y = b. y =

2

x

y

–4� 4�

x

y

3�

3�–

–1

1

c. y = d. y =

3

x

y

8�

89�

87�–

x

y

2

–2 23�

2�–



a. y = arcsin1

2b. y = arccot 0

c. y = cos−1

√2

2d. y = arcsec 2

e. y = arctan(−1) f. y = csc−1(−√

2)

g. y = cot−1(

1√3

)h. y = cos−1

(−

√3

2

)

i. y = cos(arcsin(−1)) j. y = tan

(arcsin

8

17

)


8. cos−1 0.54 9. arcsec 89

10. tan

[sin−1

(−2

5

)]11. cos(arccot 2.5)

12. Solve the equation 4 sin 3x = −2.4 for x. Assume the arc restrictions are those specified in the definition for thearcsine.



1–1x

y

2�

2�–

14. The graph below models the tides in feet, x hours after midnight on a given day.

1 6 12 18 24

1

2

3

4

5

6

Hours

a. Approximate the time (to the nearest half-hour) for low tide.

b. Approximate the time (to the nearest hour) for the second high tide.

c. Approximate the difference in the water level (to the nearest foot) between high tide and low tide.

260 Aratari, Trigonometry: Quiz 2.1 Form B

Chapter 2 Answers

Quiz 2.1 Form A

1.

x

y

2��–2� –�

2�( , 1)

23�( , –1)

1

–1

(0, 0) (2�, 0)(�, 0)

y = sin x

2.

x

y

�–2�

2�( , 0)

23�( , 0)

1

–1(�, –1)

(2�, 1)y = cos x (0, 1)

3. a. y = 7 cos x

b. y = −4 sin x

Quiz 2.1 Form B

1.

x

y

�–2�

2�( , 0)

23�( , 0)

1

–1(�, –1)

(2�, 1)y = cos x (0, 1)

Aratari, Trigonometry: Chapter 2 Answers 261

2. Range: |y| ≤ 3

x-intercepts: x = k · π

–3

4

x

y

2�–2�

y = –3 sin x

3. a. y = 4 sin x

b. y = −2 cos x

Quiz 2.1–2.2 Form A

1. a. Amplitude: 4

Period: 2π

x-intercepts: x = π

2+ k · π

4

x

y

2��–2� –�

(0, –4)

(�, 4)

(2�, –4)

23�( , 0)

2�( , 0)

b. Amplitude: 1

Period: 8π

x-intercepts: x = k · 4π

Range: −1 ≤ y ≤ 1x

y

8�4�–8� –4�

1

–1

(0, 0) (8�, 0)

(6�, –1)

(4�, 0)

(2�, 1)

2. a–b Other answers are possible

a. y = 2 sin(x + π

4

), y = 2 cos

(x − π

4

)

b. y = −6 cos

(1

2x

), y = 6 sin

[1

2(x − π)

]


Quiz 2.1–2.2 Form B

1. a. Amplitude: 2

Period: 2π

Range: |y| ≤ 2

x-intercepts: x = k · π

–3

x

y

2��–2� –�

(�, 0) (2�, 0)

23�( , –2)

2�( , 2)

(0, 0)

y = 2 sin x

b. Amplitude: 1

Period: 4π

Range: |y| ≤ 1

x-intercepts: x = π + k · 2πx

y

4�2�–4�

–1

(0, 1)

(2�, –1)

(3�, 0)

(4�, 1)

(�, 0)

y = cos( x)12

2. a–b Other answers are possible

a. y = −5 sin 4x, y = 5 sin[4(x + π

4

)]

b. y = 3 cos(x + π

4

), y = 3 sin

(x + 3π

4

)

Quiz 2.1–2.4 Form A

1. a. Period: π

Amplitude: 5

x-intercepts: x = k · π2

5

x

y

�–�

(�, 0)(0, 0)

4�( , 5)

2�( , 0)

43�( , –5)

y = 5 sin (2x)


b. Period: 2π

Amplitude: 3

Range: |y| ≤ 3

Phase Shift: rightπ

2

3

–3

x

y

2��–2�

(2�, 0)(�, 0)(0, 0)

2�( , 3)

2�(x – )

23�( , –3)

y = 3 cos

2. Period: 4π

Asymptotes: x = 2π + k · 4π

x

y

2�–2�

2

1y = tan 1

4

3. y = 4 csc 2x


1. a. Period: 8π

Amplitude: 2

x-intercepts: x = k · 4π 2

x

y

–8� 8�4�

(8�, 0)(4�, 0)

(6�, –2)

(2�, 2)

(0, 0)

y = 2 sin( x)14

b. Period: 2π

Amplitude: 4

Phase Shift: leftπ

2

4

–4

x

y

–2� 2�

(2�, 0)

(�, 0)

23�( , 4)

( , –4)

2�

2�

(x + )

(0, 0)

y = 4 cos

264 Aratari, Trigonometry: Quiz 2.5 Form A

2. Period: 2π

Asymptotes: x = k · 2π

21

x

y

–2� 2��

y = cot( x)12

3. y = 5 sec 2x

Quiz 2.5 Form A

1. a. cos y =√

2

2

y = π

4

b. tan y =√

3

3

y = π

6

c. sin y = −√

3

2

y = −π

3

d. sec y = −2

cos y = −1

2

y = 2π

3

2. a. −1.0552

b. 0.2014



1. a. x-intercepts: x = π

4+ k · π

2Asymptotes: x = k · π

2

x

y

2�

2�–

–1

y = tan �4

b. x-intercepts: none

Asymptotes: x = k · 4π

–8� 8�–1

1

x

y y = –csc x14

2. y = − cos x + 2, y = sin(x − π

2

)+ 2

3. a. cos y =√

3

2

y = π

6

b. tan y = −1

y = −π

4

c. sin y = −√

2

2

y = −π

4

d. csc y = 1

2

sin y = 2 ⇒ not possible, since sin y can not be greater than 1.

4. a. 0.5894

b. 1.2847

266 Aratari, Trigonometry: Chapter 2 Test Form A

Chapter 2 Test Form A

1.

–4

4

x

y

�–�

y = 4 cos(–2x) Period π

Amplitude 4

Range |y| ≤ 4

2.

–1

1

x

y

4�–4�

y = sin[ (x + �)]12

Period 4π

Amplitude 1

Phase Shift left π

x-intercepts x = π + k · 2π

3.

–1

1

x

y

4�

4�–

y = –cot(4x)

Periodπ

4x-intercepts x = π

8+ k · π

4

Asymptotes x = k · π4

4.

1

x

y

2�

–2�2

3�2�

23�

2�––

(x – )y = 4 csc

Period 2π

Range |y| ≥ 4

Aratari, Trigonometry: Chapter 2 Test Answers 267

5.

3

x

y

2 4 6 8

2�( x)y = cos + 2

Period 4

Range 1 ≤ y ≤ 3

6. a–d Other answers are possible

a. y = −2 sin 4x b. y = − cot

(1

4x

)

c. y = 2 sec x d. y = 3 cos(x − π

4

)7. a. y = π

2b. y = −π

4

c. y = π

6d. y = π

6

e. y = π

3f. y = −π

4

g. y = 2π

3h. y = −π

2

i. y = −1

2j. y = 8

15

8. −0.4668 9. 1.5520

10. −0.5686 11. 2.6926

12. x = 1

4cos−1

(2

7

)≈ 0.3203

13. y = arcsin x

14. a. 1 day, 31

2days

b. 21

2days

c. 6◦


Chapter 2 Test Form B

1.

–1

3

x

y

8�–8�

y = 3 sin 14

Period 8π

Amplitude 3

Range |y| ≤ 3

2.

�–� x

y

–1

y = cos[2(x + )]�4

Period π

Amplitude 1

Phase Shift leftπ

4x-intercepts x = k · π

2

3.

x

y

–3

–1

1

29�

23�

23�–

y = –tan( x)13

Period 3π

x-intercepts x = k · 3π

Asymptotes x = 3π

2+ k · 3π

4.

x

y

2�–2�2�–

y = 4 sec x

–3

1

4

–4

Period 2π

Range |y| ≥ 4


5.

1

–3

x

y

3�

y = sin – 1

3 12

Period 6

Range −2 ≤ y ≤ 0


a. y = cot 3x b. y = 4 sin(x − π

6

)

c. y = −2 csc

(1

2x

)d. y = sec x + 3

7. a. y = π

2b. y = π

6

c. y = π

3d. y = π

3

e. y = −π

4f. y = 3π

4

g. y = −π

6h. y = π

i. y =√

3

2j. y = 8

17

8. 2.0376 9. 1.5596

10. 2.1394 11. 0.4000

12. x = 1

4sin−1

(5

7

)≈ 0.1989

13. y = arccos x

14. a. 1 day, 3 days

b. 2 days

c. 6◦


Chapter 2 Test Form C

1.

–1x

y

–8� 8�4�

y = 3 cos(– x)14

Period 8π

Amplitude 3

Range |y| ≤ 3

2.

x

y

–� �

y = sin[2(x + )]�4

Period π

Amplitude 1

Phase Shift leftπ

4

x-intercepts x = π

4+ k · π

2

3.

x

y

3�–3�

y = –cot ( x)13

Period 3π

x-intercepts x = 3π

2+ k · 3π

Asymptotes x = k · 3π

4.

�–2� 2� x

y

4

2

y = 4 csc x

Period 2π

Range |y| ≥ 4

5.1

–1x

y

6 12

3�( x)y = cos – 1

Period 6

Range −2 ≤ y ≤ 0



a. y = −4 sin

(1

3x

)b. y = 1

2sec(2x)

c. y = tan

[1

2

(x − 5π

6

)]d. y = sin x + 2

7. a. y = π

3b. y = 0

c. y = π

4d. y = π

6

e. y = π f. y = −π

4

g. y = π

3h. y = −π

2

i. y =√

2

2j. y = 15

17

8. 0.5704 9. 0.0112

10. 0.4741 11. 2.2913

12. x = 1

5cos−1

(2

3

)≈ 0.1682

13. y = arctan x

14. Answers can be slightly off due to the quality of the graph.

14. a. 2:30 am, 3 pm

b. 8:30 am, 9:30 pm

c. 1 ft


Chapter 2 Test Form D

1.

–4

x

y

�–�

y = 4 sin (–2x)Period π

Amplitude 4

Range |y| ≤ 4

2.

x

y

4��–4�

1

–1

y = cos[ (x – �)]12

Period 4π

Amplitude 1

Phase Shift right π

x-intercepts x = k · 2π

3.

x

y

–3

–1

1

83�

8�–

8�

y = –tan (4x)

Periodπ

4x-intercepts x = k · π

4

Asymptotes x = π

8+ k · π

4

4.

x

y

2�–2� –�

y = 4 sec

2

–2

(x + )�2

Period 2π

Range |y| ≥ 4


5.

1

–1x

y

1 2 4

y = sin(�x) + 1

Period 2

Range 0 ≤ y ≤ 2


a. y = −2 cos

(1

2x

)b. y = − cot(3x)

c. y = 3 sin(x − π

8

)d. y = 2 sec x − 2

7. a. y = π

6b. y = π

2

c. y = π

4d. y = π

3

e. y = −π

4f. y = −π

4

g. y = π

3h. y = 5π

6

i. y = 0 j. y = 8

15

8. 1.0004 9. 1.5596

10. −0.4364 11. 0.9285

12. x = 1

3arcsin

(−2.4

4

)≈ −0.2145

13. y = arcsin x

14. Answers can be slightly off due to the quality of the graph.

14. a. 10:30 am

b. 6 pm

c. 4 ft

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