2.R: Periodic Functions (Review)

6.1: Graphs of the Sine and Cosine Functions

For the exercises 1-8, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

1) $f(x)=-3\cos x+3$

Answer

amplitude: $3$; period: $2\pi$; midline: $y=3$; no asymptotes

2) $f(x)=\dfrac{1}{4}\sin x$

3) $f(x)=3\cos\left ( x+\dfrac{\pi }{6} \right )$

Answer

amplitude: $3$; period: $2\pi$; midline: $y=0$; no asymptotes

4) $f(x)=-2\sin\left ( x-\dfrac{2\pi }{3} \right )$

5) $f(x)=3\sin\left ( x-\dfrac{\pi }{4} \right )-4$

Answer

amplitude: $3$; period: $2\pi$; midline: $y=-4$; no asymptotes

6) $f(x)=2\left (\cos\left ( x-\dfrac{4\pi }{3} \right )+1 \right )$

7) $f(x)=6\sin\left ( 3x-\dfrac{\pi }{6} \right )-1$

Answer

amplitude: $6$; period: $dfrac{2\pi }{3}$; midline: $y=-1$; no asymptotes

8) $f(x)=-100\sin(50x-20)$

6.2: Graphs of the Other Trigonometric Functions

For the exercises 1-4, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

1) $f(x)=\tan x-4$

Answer

stretching factor: none; period: $\pi$; midline: $y=-4$; asymptotes: $x=\dfrac{\pi }{2}+\pi k$, where $k$ is an integer

2) $f(x)=2\tan \left ( x-\dfrac{\pi }{6} \right )$

3) $f(x)=-3\tan (4x)-2$

Answer

stretching factor: $3$; period: $\dfrac{\pi }{4}$; midline: $y=-2$; asymptotes: $x=\dfrac{\pi }{8}+\dfrac{\pi }{4}k$, where $k$ is an integer

4) $f(x)=0.2\cos(0.1x)+0.3$

For the exercises 5-10, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.

5) $f(x)=\dfrac{1}{3}\sec x$

Answer

amplitude: none; period: $2\pi$; no phase shift; asymptotes: $x=\dfrac{\pi }{2}k$, where $k$ is an integer

6) $f(x)=3\cot x$

7) $f(x)=4\csc (5x)$

Answer

amplitude: none; period: $\dfrac{2\pi }{5}$; no phase shift; asymptotes: $x=\dfrac{\pi }{5}k$, where $k$ is an integer

8) $f(x)=8\sec \left (\dfrac{1}{4}x \right )$

9) $f(x)=\dfrac{2}{3}\csc \left (\dfrac{1}{2}x \right )$

Answer

amplitude: none; period: $4\pi$; no phase shift; asymptotes: $x=2\pi k$, where $k$ is an integer

10) $f(x)=-\csc (2x+\pi)$

For the exercises 11-15, use this scenario: The population of a city has risen and fallen over a $20$-year interval. Its population may be modeled by the following function: $y=12,000+8,000\sin(0.628x)$, where the domain is the years since 1980 and the range is the population of the city.

11) What is the largest and smallest population the city may have?

Answer

largest: $20,000$; smallest: $4,000$

12) Graph the function on the domain of $[0,40]$.

13) What are the amplitude, period, and phase shift for the function?

Answer

amplitude: $8,000$; period: $10$; phase shift: $0$

14) Over this domain, when does the population reach $18,000$? $13,000$?

15) What is the predicted population in 2007? 2010?

Answer

In 2007, the predicted population is $4,413$. In 2010, the population will be $11,924$.

For the exercises 16a-16d, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

16) Suppose the graph of the displacement function is shown in the Figure below, where the values on the $x$-axis represent the time in seconds and the $y$-axis represents the displacement in inches.

1. Give the equation that models the vertical displacement of the weight on the spring.
2. At $\text{time} = 0$, what is the displacement of the weight?

Answer

$5$ in.

3. At what time does the displacement from the equilibrium point equal zero?
4. What is the time required for the weight to return to its initial height of $5$ inches? In other words, what is the period for the displacement function?

Answer

$10$ seconds

6.3: Inverse Trigonometric Functions

For the exercises 1-11, find the exact value without the aid of a calculator.

1) $\sin ^{-1}(1)$

2) $\cos ^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )$

Answer

$\dfrac{\pi }{6}$

3) $\tan ^{-1}(-1)$

4) $\cos ^{-1}\left ( \dfrac{1}{\sqrt{2}} \right )$

Answer

$\dfrac{\pi }{4}$

5) $\sin ^{-1}\left ( \dfrac{-\sqrt{3}}{2} \right )$

6) $\sin ^{-1}\left (\cos \left (\dfrac{\pi }{6} \right ) \right )$

Answer

$\dfrac{\pi }{3}$

7) $\cos ^{-1}\left (\tan \left (\dfrac{3\pi }{4} \right ) \right )$

8) $\sin \left (\sec^{-1} \left (\dfrac{3}{5} \right ) \right )$

Answer

No solution

9) $\cot \left (\sin^{-1} \left (\dfrac{3}{5} \right ) \right )$

10) $\tan \left (\cos^{-1} \left (\dfrac{5}{13} \right ) \right )$

Answer

$\dfrac{12}{5}$

11) $\sin \left (\cos^{-1} \left (\dfrac{x}{x+1} \right ) \right )$

12) Graph $f(x)=\cos x$ and $f(x)=\sec x$ on the interval $[0,2\pi )$ and explain any observations.

Answer

The graphs are not symmetrical with respect to the line $y=x$. They are symmetrical with respect to the $y$-axis.

13) Graph $f(x)=\sin x$ and $f(x)=\csc x$ and explain any observations.

14) Graph the function $f(x)=\dfrac{x}{1}-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}$ on the interval $[-1,1]$ and compare the graph to the graph of $f(x)=\sin x$ on the same interval. Describe any observations.

Answer

The graphs appear to be identical.

Practice Test

For the exercises 1-13, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

1) $f(x)=0.5\sin x$

Answer

amplitude: $0.5$; period: $2\pi$; midline $y=0$

 y=0  y=0  y=0 

2) $f(x)=5\cos x$

3) $f(x)=5\sin x$

Answer

amplitude: $0.5$; period: $2\pi$; midline $y=0$

4) $f(x)=\sin (3x)$

5) $f(x)=-\cos \left ( x+\dfrac{\pi }{3} \right )+1$

Answer

amplitude: $1$; period: $2\pi$; midline $y=1$

6) $f(x)=5\sin \left (3\left ( x-\dfrac{\pi }{6} \right ) \right )+4$

7) $f(x)=3\cos \left ( \dfrac{1}{3}x-\dfrac{5\pi }{6} \right )$

Answer

amplitude: $3$; period: $6\pi$; midline $y=0$

8) $f(x)=\tan (4x)$

9) $f(x)=-2\tan \left ( x-\dfrac{7\pi }{6} \right )+2$

Answer

amplitude: none; period: $\pi$; midline $y=0$, asymptotes: $x=\dfrac{2\pi }{3}+\pi k$, where $k$ is an integer

10) $f(x)=\pi \cos(3x+\pi)$

11) $f(x)=5\csc(3x)$

Answer

amplitude: none; period: $\dfrac{2\pi }{3}$; midline $y=0$, asymptotes: $x=\dfrac{\pi }{3}k$, where $k$ is an integer

12) $f(x)=\pi \sec \left ( \dfrac{\pi }{2}x \right )$

13) $f(x)=2\csc \left ( x+\dfrac{\pi }{4} \right )-3$

Answer

amplitude: none; period: $2\pi$; midline $y=-3$

For the exercises 14-16, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

14) Give in terms of a sine function.

15) Give in terms of a sine function.

Answer

amplitude: $2$; period: $2$; midline: $y=0$; $f(x)=2\sin(\pi (x-1))$

16) Give in terms of a tangent function.

For the exercises 17-20, find the amplitude, period, phase shift, and midline.

17) $y=\sin\left(\dfrac{\pi}{6}x+\pi \right)-3$

Answer

amplitude: $1$; period: $12$; phase shift: $-6$; midline: $y=-3$

18) $y=8\sin\left(\dfrac{7\pi}{6}x+\dfrac{7\pi}{2} \right)+6$

19) The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is $68^{\circ}$F at midnight and the high and low temperatures during the day are $80^{\circ}$F and $56^{\circ}$F, respectively. Assuming $t$ is the number of hours since midnight, find a function for the temperature, $D$, in terms of $t$.

Answer

$D(t)=68-12\sin\left(\dfrac{\pi}{12}x \right)$

20) Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is $3$ feet at its lowest at 2:00 a.m. and $71$ feet at its highest, which occurs every $5$ hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

For the exercises 21-25, find the period and horizontal shift of each function.

21) $g(x)=3\tan(6x+42)$

Answer

period: $\dfrac{\pi}{6}$; horizontal shift: $-7$

22) $n(x)=4\csc \left(\dfrac{5\pi }{3}x-\dfrac{20\pi }{3} \right)$

23) Write the equation for the graph in the Figure below in terms of the secant function and give the period and phase shift.

Answer

$f(x)=\sec(\pi x)$; period: $2$; phase shift: $0$

24) If $\tan x=3$, find $\tan (-x)$.

25) If $\sec x=4$,   find $\sec (-x)$.

Answer

$4$

For the exercises 26-28, graph the functions on the specified window and answer the questions.

26) Graph $m(x)=\sin(2x)+\cos(3x)$ on the viewing window $[-10,10]$ by $[-3,3]$. Approximate the graph’s period.

27) Graph $n(x)=0.02\sin(50\pi x)$ on the following domains in $x:[0,1]$ and$[0,3]$. Suppose this function models sound waves. Why would these views look so different?

Answer

The views are different because the period of the wave is $125$. Over a bigger domain, there will be more cycles of the graph.

28) Graph $f(x)=\dfrac{\sin x}{x}$ on $[-0.5,0.5]$ and explain any observations.

For the exercises 29-31, let $f(x)=\dfrac{3}{5}\cos(6x)$.

29) What is the largest possible value for $f(x)$?

Answer

$\dfrac{3}{5}$

30) What is the smallest possible value for $f(x)$?

31) Where is the function increasing on the interval $[0,2\pi ]$?

Answer

On the approximate intervals $(0.5,1),(1.6,2.1),(2.6,3.1),(3.7,4.2),(4.7,5.2),(5.6,6.28)$

For the exercises 32-33, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

32) Sine curve with amplitude $3$, period $\dfrac{\pi }{3}$, and phase shift $(h,k)=\left(\dfrac{\pi }{4},2\right)$

33) Cosine curve with amplitude $2$, period \(\dfrac{\pi }{6}\), and phase shift $(h,k)=\left(-\dfrac{\pi }{4},3\right)$

Answer

$f(x)=2\cos\left ( 12\left ( x+\dfrac{\pi }{4} \right ) \right )+3$

For the exercises 34-35, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

34) $f(x)=5\cos(3x)+4\sin(2x)$

35) $f(x)=e^{(sint)}$

Answer

This graph is periodic with a period of $2\pi$

For the exercises 36-43, find the exact value.

36) $\sin^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )$

37) $\tan^{-1}\left ( \sqrt{3} \right )$

Answer

$\dfrac{\pi }{3}$

38) $\cos^{-1}\left ( -\dfrac{\sqrt{3}}{2} \right )$

39) $\cos^{-1}\left ( \sin(\pi) \right )$

Answer

$\dfrac{\pi }{2}$

40) $\cos^{-1}\left ( \tan \left (\dfrac{7\pi}{4} \right ) \right )$

41) $\cos(\sin^{-1}(1-2x))$

Answer

$\sqrt{1-(1-2x)^2}$

42) $\cos^{-1}(-0.4)$

43) $\cos \left (\tan^{-1}\left(x^2\right) \right )$

Answer

$\dfrac{1}{\sqrt{1+x^4}}$

For the exercises 44-46, suppose $\sin t=\dfrac{x}{x+1}$ . Evaluate the following expressions.

44) $\tan t$

45) $csc t$

Answer

$\dfrac{x+1}{x}$

46) Given Figure, find the measure of angle $\theta$ to three decimal places. Answer in radians.

For the exercises 47-49, determine whether the equation is true or false.

47) $\arcsin\left(\sin\left(\dfrac{5\pi }{6}\right)\right)=\dfrac{5\pi }{6}$

Answer

False

48) $\arccos\left(\cos\left(\dfrac{5\pi }{6}\right)\right)=\dfrac{5\pi }{6}$

49) The grade of a road is $7\%$. This means that for every horizontal distance of $100$ feet on the road, the vertical rise is $7$ feet. Find the angle the road makes with the horizontal in radians.

Answer

approximately $0.07$ radians