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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
Chapter 5 Radical Expressions and Equations
5. 5 6. 5. 5. 5. 5. 5. 2 160. 2 2 (5)( )( )( ). 4. 5
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MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 10 of 75 b) The pulse rate for this person is 60 0 8 or 75 beats per minute Section 5 1 Page 235 Question 16 Examples: Step 1 Step 2 Angle x Opposite (cm) Hypotenuse (cm) sin x = opposite hypotenuse 0°
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What is the complete solution of 5 5?
- Add 5 5 to both sides of the equation. The complete solution is the result of both the positive and negative portions of the solution. The result can be shown in multiple forms. x = 8.46410161…,1.53589838… x = 8.46410161 …, 1.53589838 …
What is the purpose of Chapter 5 in physics?
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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, amaximum, 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 - 32, -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 75Section 5.1 Page 233 Question 5
a)For the function y = sin 4ș, b = 4.
360Period||
360904
b
2ʌPeriod||
2ʌ 24bThe period is 90° or
2. b)For the function y = cos
13ș, b = 1
3.360Period||
1 3601080
3b
2ʌPeriod|
3| 2ʌ6ʌ1b
The period is 1080° or 6ʌ.
c) For the function y = sin 23x, b = 23.
360Period||
3600 2 54
3b
2ʌPeriod|
3| 2ʌ 3 2bThe period is 540° or 3
d)For the function y = cos 6x, b = 6.
360Period||
360606b
2ʌPeriod||
2ʌ 36bThe period is 60° or
3. MHR • 978-0-07-0738850 Pre-Calculus 12 Solutions Chapter 5 Page 4 of 75Section 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 23: 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 23. The graph of
this function is related to the graph of y = sin x by a vertical stretch by a factor of 23 and a
reflection in the x-axis.Section 5.1 Page 234 Question 8
a)For the function y = cos 2x, b = 2.
360Period||
36082
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||
360103
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
14x, b = 1
4.360Period||
1 3601440
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
23x, b = 23.
360Period||
3600 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||
36061
30b
2ʌPeriod||
2ʌ2ʌ1b
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||
36082
10b
2ʌPeriod|
2| 2ʌ ʌb c)For the function y =
53 sin 2
3 x , a = 5 3 and b = - 23. The amplitude is 5
3, or 5
3.360Period||
3 3 40260
5 b
2ʌPeriod|
2| 2ʌ 3 3 b d)For the function y = 3 sin
1 2 x, a = 3 and b = 12. The amplitude is |3|, or 3.
360Period||
3600 1 72
2 b
2ʌPeriod|
2| 2ʌ4ʌ1
bSection 5.1 Page 234 Question 10
a)Use Amplitude =
maximum value minimum value 2The 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 7542ʌP
ʌeriod
2ʌ 1 2 b b bSo, 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 positive2ʌPeriod||
2ʌ 2 b b bSo, 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
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