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Mathematics LibreTexts

6.1E: Exercises

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6.1: Graphs of the Sine and Cosine Functions

In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions

Verbal

1) Why are the sine and cosine functions called periodic functions?

Answer

The sine and cosine functions have the property that f(x+P)=f(x) for a certain P. This means that the function values repeat for every P units on the x-axis.

2) How does the graph of y=sinx compare with the graph of y=cosx? Explain how you could horizontally translate the graph of y=sinx to obtain y=cosx.

3) For the equation Acos(Bx+C)+Dwhat constants affect the range of the function and how do they affect the range?

Answer

The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically.

4) How does the range of a translated sine function relate to the equation y=Asin(Bx+C)+D

5) How can the unit circle be used to construct the graph of f(t)=sint?

Answer

At the point where the terminal side of t intersects the unit circle, you can determine that the sint equals the y-coordinate of the point.

Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x>0. Round answers to two decimal places if necessary.

6) f(x)=2sinx

7) f(x)=23cosx

Answer

Ex 6.1.7.png

amplitude: 23period: 2πmidline: y=0maximum: y=23 occurs at x=0minimum: y=23 occurs at x=πfor one period, the graph starts at 0 and ends at 2π.

8) f(x)=3sinx

Exercise 6.1E.9

f(x)=4sinx

Answer

Ex 6.1.9.png

amplitude: 4; period: 2πmidline: y=0maximum y=4 occurs at x=π2minimum: y=4 occurs at x=3π2one full period occurs from x=0 to x=2π

10) f(x)=2cosx

Exercise 6.1E.11

f(x)=cos(2x)

Answer

Ex 6.1.11.png

amplitude: 1; period: πmidline:

12) f(x)=2sin(12x)

13) f(x)=4cos(πx)

Answer

Ex 6.1.13.png

amplitude: 4; period: 2; midline: y=0maximum: y=4 occurs at x=0minimum: y=4 occurs at x=1

14) f(x)=3cos(65x)

15) y=3sin(8(x+4))+5

Answer

CNX_Precalc_Figure_06_01_210.jpg

amplitude: 3; period: π2 midline: y=5 maximum: y=8 occurs at x=0.12 minimum: y=2 occurs at x=0.516 horizontal shift: 4 vertical translation 5; one period occurs from x=0 to x=π4

16) y=2sin(3x21)+4

17) y=5sin(5x+20)2

Answer

CNX_Precalc_Figure_06_01_212.jpg

amplitude: 5; period:2π5 midline: y=2 maximum: y=3 occurs at x=0.08 minimum: y=7 occurs at x=0.71 phase shift: 4 vertical translation: 2 one full period can be graphed on x=0 tox=2π5

For the following exercises, graph one full period of each function, starting at x=0. For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x>0. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

18) f(t)=2sin(t5π6)

19) f(t)=cos(t+π3)+1

Answer

CNX_Precalc_Figure_06_01_214.jpg

amplitude: 1; period: 2π midline: y=1 maximum: y=2 occurs at x=2.09 maximum: y=2 occurs at t=2.09 minimum: y=0 occurs at t=5.24 phase shift: π3 vertical translation: 1; one full period is from t=0 to t=2π

20) f(t)=4cos(2(t+π4))3

21) f(t)=sin(12t+5π3)

Answer

CNX_Precalc_Figure_06_01_216.jpg

amplitude: 1; period: 4π midline: y=0 maximum: y=1 occurs at t=11.52 minimum: y=1 occurs at t=5.24 phase shift: 10π3 vertical shift: 0

22) f(x)=4sin(π2(x3))+7

Exercise 6.1E.23

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure below.

CNX_Precalc_Figure_06_01_218.jpg

Answer

amplitude: 2; midline: y=3 ;   period: 4; equation: f(x)=2sin(π2x)3

24) Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in the Figure below.

CNX_Precalc_Figure_06_01_219.jpg

25) Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in the Figure below.

CNX_Precalc_Figure_06_01_220.jpg

Answer

amplitude: 2; period: 5; midline: y=3 equation: f(x)=2cos(2π5x)+3

26) Determine the amplitude, period, midline, and an equation involving sine for the graph shown in the Figure below.

CNX_Precalc_Figure_06_01_221.jpg

27) Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in the Figure below.

CNX_Precalc_Figure_06_01_222.jpg

Answer

amplitude: 4; period: 2; midline: y=0 equation: f(x)=4cos(π(xπ2))

28) Determine the amplitude, period, midline, and an equation involving sine for the graph shown in the Figure below.

CNX_Precalc_Figure_06_01_223.jpg

29) Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in the Figure below.

CNX_Precalc_Figure_06_01_224.jpg

Answer

amplitude: 2; period: 2; midline y=1 equation: f(x)=2cos(πx)+1

30) Determine the amplitude, period, midline, and an equation involving sine for the graph shown in the Figure below.

Algebraic

For the following exercises, let f(x)=sinx

31) On [0,2π) solve f(x)=0

32) On [0,2π) , solve f(x)=12.

Answer

π6, 5π6

33) Evaluate f(π2)

34) On [0,2π), f(x)=22

Answer

π4, 3π4

35) On [0,2π) the maximum value(s) of the function occur(s) at what x-value(s)?

36) On [0,2π) the minimum value(s) of the function occur(s) at what x-value(s)?

Answer

3π2

37) Show that f(x)=f(x) This means that f(x)=sinx is an odd function and possesses symmetry with respect to ________________.

For the following exercises, let f(x)=cosx

38) On [0,2π) solve the equation f(x)=cosx=0

Answer

π2, 3π2

39) On [0,2π) solve f(x)=12.

40) On [0,2π) find the x-intercepts of f(x)=cosx.

Answer

π2, 3π2

41) On [0,2π) find the x-values at which the function has a maximum or minimum value.

42) On [0,2π) solve the equation f(x)=32

Answer

π6, 11π6

Technology

43) Graph h(x)=x+sinx on [0,2π] Explain why the graph appears as it does.

44) Graph h(x)=x+sinx on [100,100] Did the graph appear as predicted in the previous exercise?

Answer

The graph appears linear. The linear functions dominate the shape of the graph for large values of x.

CNX_Precalc_Figure_06_01_227.jpg

45) Graph f(x)=xsinx on [0,2π] and verbalize how the graph varies from the graph of f(x)=xsinx.

46) Graph f(x)=xsinx on the window [10,10] and explain what the graph shows.

Answer

The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.

CNX_Precalc_Figure_06_01_229.jpg

47) Graph f(x)=sinxx on the window [5π,5π] and explain what the graph shows.

Real-World Applications

48) A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.

  1. Find the amplitude, midline, and period of h(t).
  2. Find a formula for the height function h(t).
  3. How high off the ground is a person after 5 minutes?
Answer
  1. Amplitude: 12.5; period: 10; midline: y=13.5
  2. h(t)=12.5sin(π5(t2.5))+13.5
  3. 26 ft

6.1E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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