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MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 1 of 75

Chapter 5 Trigonometric Functions Graphs

Section 5.1 Graphing Sine and Cosine Functions

Section 5.1

Page 233 Question 1

a) One cycle of the sine function y = sin x, from 0 to 2ʌ, includes three x-intercepts, a

maximum, and a minimum. These five key points divide the period into quarters: (0, 0), ʌ,12, (ʌ, 0), 3ʌ,1

2, and (2ʌ, 0).

b) c) The x-intercepts of the graph of y = sin x for -2ʌ x 2ʌ are -2ʌ, -ʌ, 0, ʌ, and 2ʌ. d)

The y-intercept of the graph of y = sin x

is 0. e) For the graph of y = sin x, the maximum value is 1 and the minimum value is -1.

Section 5.1 Page 233 Question 2

a) One cycle of the function y = cos x, from 0 to 2ʌ , includes two x-intercepts, two maximums, and a minimum. These five key points divide the period into quarters: (0, 1),

ʌ,02, (ʌ, -1), 3ʌ,02

, and (2ʌ, 1). b) c) The x-intercepts of the graph of y = cos x for -2ʌ x 2ʌ are - 3

2, -ʌ

2, ʌ

2, and 3ʌ

2. MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 2 of 75 d) The y-intercept of the graph of y = cos x is 1. e) For the graph of y = cos x, the maximum value is 1 and the minimum value is -1.

Section 5.1 Page 233 Question 3

Section 5.1 Page 233 Question 4

a)

For the function y = 2 sin ș, a = 2. The

amplitude is |2|, or 2. b) For the function y = 1 2 cos x, a = 1 2.

The amplitude is

1 2 , or 1 2 c)

For the function y = -

1 3 sin x, a = - 1 3.

The amplitude is

1 3 , or 1 3 d)

For the function y = -6 cos ș, a = -6.

The amplitude is |-6|, or 6.

MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 3 of 75

Section 5.1 Page 233 Question 5

a)

For the function y = sin 4ș, b = 4.

360Period||

360
904
b

2ʌPeriod||

2ʌ 24b

The period is 90° or

2. b)

For the function y = cos

1

3ș, b = 1

3.

360Period||

1 360
1080
3b

2ʌPeriod|

3| 2ʌ

6ʌ1b

The period is 1080° or 6ʌ.

c) For the function y = sin 2

3x, b = 23.

360Period||

360
0 2 54
3b

2ʌPeriod|

3| 2ʌ 3 2b

The period is 540° or 3

d)

For the function y = cos 6x, b = 6.

360Period||

360
606b

2ʌPeriod||

2ʌ 36b

The period is 60° or

3. MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 4 of 75

Section 5.1 Page 233 Question 6

a) For the function y = 3 cos x, a = 3 and b = 1. The graph of this cosine function will have an amplitude of 3 and a period of 2ʌ: choice A. b) For the function y = cos 3x, a = 1 and b = 3. The graph of this cosine function will have an amplitude of 1 and a period of 2

3: choice

D. c) For the function y = -sin x, a = -1 and b = 1. The graph of this sine function will have an amplitude of 1, be reflected in the x-axis, and have a period of 2ʌ: choice C. d) For the function y = -cos x, a = -1 and b = 1. The graph of this cosine function will have an amplitude of 1, be reflected in the x-axis, and have a period of 2ʌ: choice B.

Section 5.1 Page 234 Question 7

a) For the function y = 3 sin x, a = 3. The amplitude is |3|, or 3. The graph of this function is related to the graph of y = sin x by a vertical stretch by a factor of 3. b) For the function y = -5 sin x, a = -5. The amplitude is |-5|, or 5. The graph of this function is related to the graph of y = sin x by a vertical stretch by a factor of 5 and a reflection in the x-axis. c) For the function y = 0.15 sin x, a = 0.15. The amplitude is |0.15|, or 0.15. The graph of this function is related to the graph of y = sin x by a vertical stretch by a factor of 0.15. d) For the function y = 2 3 sin x, a = 2 3 . The amplitude is 2 3 , or 2

3. The graph of

this function is related to the graph of y = sin x by a vertical stretch by a factor of 2

3 and a

reflection in the x-axis.

Section 5.1 Page 234 Question 8

a)

For the function y = cos 2x, b = 2.

360Period||

360
82
10b The graph of this function is related to the graph of y = cos x by a horizontal stretch by a factor of 1 2. MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 5 of 75 b) For the function y = cos (-3x), b = -3.

360Period||

360
103
2b The graph of this function is related to the graph of y = cos x by a horizontal stretch by a factor of 1

3 and a reflection in the y-axis.

c)

For the function y = cos

1

4x, b = 1

4.

360Period||

1 360
1440
4b The graph of this function is related to the graph of y = cos x by a horizontal stretch by a factor of 4. d)

For the function y = cos

2

3x, b = 23.

360Period||

360
0 2 54
3b The graph of this function is related to the graph of y = cos x by a horizontal stretch by a factor of 3 2.

Section 5.1 Page 234 Question 9

a) For the function y = 2 sin x, a = 2 and b = 1. The amplitude is |2|, or 2.

360Period||

360
61
30b

2ʌPeriod||

2ʌ

2ʌ1b

MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 6 of 75 b) For the function y = -4 sin 2x, a = -4 and b = 2. The amplitude is |-4|, or 4.

360Period||

360
82
10b

2ʌPeriod|

2| 2ʌ ʌb c)

For the function y =

5

3 sin 2

3 x , a = 5 3 and b = - 2

3. The amplitude is 5

3, or 5

3.

360Period||

3 3 402
60
5 b

2ʌPeriod|

2| 2ʌ 3 3 b d)

For the function y = 3 sin

1 2 x, a = 3 and b = 1

2. The amplitude is |3|, or 3.

360Period||

360
0 1 72
2 b

2ʌPeriod|

2| 2ʌ

4ʌ1

b

Section 5.1 Page 234 Question 10

a)

Use Amplitude =

maximum value minimum value 2

The amplitude of graph A is

2() 22
, or 1, and the period is 4ʌ.

The amplitude of graph B is

(.5) 200.5
, or 0.5, and the period is ʌ. b) Graph A has the pattern of a sine curve. Since the amplitude is 1, a = 1. Using the period of 4

ʌ and choosing

b to be positive MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 7 of 75

42ʌP

ʌeriod

2ʌ 1 2 b b b

So, the equation of the function in the form

y = a sin bx is y = sin 1 2 x. Graph B has the pattern of a cosine curve. Since the amplitude is 0.5, a = 0.5. Using the period of and choosing b to be positive

2ʌPeriod||

2ʌ 2 b b b

So, the equation of the function in the form

y = a cos bx is y = 0.5 cos 2x. c) Since graph passes through (0, 0), the sine function is the better choice. Since graph B passes through (0, 1), the cosine function is the better choice.

Section 5.1 Page 234 Question 11

a) For y = 2 cos x in the interval [-360°, 360°]: b) For y = -3 sin x in the interval [-360°, 360°]: MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 8 of 75 c) For y = 1 2 sin x in the interval [-360°, 360°]: d) For y = - 3 4 cos x in the interval [-360°, 360°]:

Section 5.1 Page 234 Question 12

a) Given y = 3 sin 2x and point A has coordinates (0, 0), find thequotesdbs_dbs21.pdfusesText_27