College Algebra & Trigonometry 5th Edition Chapter 8 Practice Questions

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8.1 Graphs of the Sine and Cosine Functions

5.

midline: $y=0;$ amplitude: $\left|A\right|=\frac{1}{2};$ period: $P=\frac{2\pi }{\left|B\right|}=6\pi ;$ phase shift: $\frac{C}{B}=\pi$

6.

$f\left(x\right)=\sin (x)+2$

7.

two possibilities: $y=4\sin \left(\frac{\pi }{5}x-\frac{\pi }{5}\right)+4$ or $y=-4\sin \left(\frac{\pi }{5}x+\frac{4\pi }{5}\right)+4$

8.

midline: $y=0;$ amplitude: $\left|A\right|=0.8;$ period: $P=\frac{2\pi }{\left|B\right|}=\pi ;$ phase shift: $\frac{C}{B}=0$ or none

9.

midline: $y=0;$ amplitude: $\left|A\right|=2;$ period: $P=\frac{2\pi }{\left|B\right|}=6;$ phase shift: $\frac{C}{B}=-\frac{1}{2}$

11.

$y=3\cos \left(x\right)-4$

8.2 Graphs of the Other Trigonometric Functions

2.

It would be reflected across the line $y=-1,$ becoming an increasing function.

3.

$g(x)=4\tan (2x)$

4.

This is a vertical reflection of the preceding graph because $A$ is negative.

8.3 Inverse Trigonometric Functions

1.

$\arccos (0.8776)\approx 0.5$

2.

1. ⓐ $-\frac{\pi }{2};$
2. ⓑ $-\frac{\pi }{4};$
3. ⓒ $\pi ;$
4. ⓓ $\frac{\pi }{3}$

4.

${\sin }^{\mathrm{-1}}(0.6)=\mathrm{36.87°}=0.6435$ radians

9.

$\frac{4x}{\sqrt{16x^2+1}}$

8.1 Section Exercises

1.

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

3.

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

5.

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

7.

amplitude: $\frac{2}{3};$ period: $2\pi ;$ midline: $y=0;$ maximum: $y=\frac{2}{3}$ occurs at $x=0;$ minimum: $y=-\frac{2}{3}$ occurs at $x=\pi ;$ for one period, the graph starts at 0 and ends at $2\pi$

9.

amplitude: 4; period: $2\pi ;$ midline: $y=0;$ maximum $y=4$ occurs at $x=\frac{\pi }{2};$ minimum: $y=-4$ occurs at $x=\frac{3\pi }{2};$ one full period occurs from $x=0$ to $x=2\pi$

11.

amplitude: 1; period: $\pi ;$ midline: $y=0;$ maximum: $y=1$ occurs at $x=\pi ;$ minimum: $y=-1$ occurs at $x=\frac{\pi }{2};$ one full period is graphed from $x=0$ to $x=\pi$

13.

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

15.

amplitude: 3; period: $\frac{\pi }{4};$ 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 $x=\frac{\pi }{4}$

17.

amplitude: 5; period: $\frac{2\pi }{5};$ midline: $y=\mathrm{-2};$ maximum: $y=3$ occurs at $x=0.08;$ minimum: $y=\mathrm{-7}$ occurs at $x=\mathrm{0.71;}$ phase shift: $\mathrm{-4};$ vertical translation: $\mathrm{-2;}$ one full period can be graphed on $x=0$ to $x=\frac{2\pi }{5}$

19.

amplitude: 1 ; period: $2\pi ;$ 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: $-\frac{\pi }{3};$ vertical translation: 1; one full period is from $t=0$ to $t=2\pi$

21.

amplitude: 1; period: $4\pi ;$ midline: $y=0;$ maximum: $y=1$ occurs at $t=11.52;$ minimum: $y=-1$ occurs at $t=5.24;$ phase shift: $-\frac{10\pi }{3};$ vertical shift: 0

23.

amplitude: 2; midline: $y=-3;$ period: 4; equation: $f(x)=2\sin \left(\frac{\pi }{2}x\right)-3$

25.

amplitude: 2; period: 5; midline: $y=3;$ equation: $f(x)=-2\cos \left(\frac{2\pi }{5}x\right)+3$

27.

amplitude: 4; period: 2; midline: $y=0;$ equation: $f(x)=-4\cos \left(\pi \left(x-\frac{\pi }{2}\right)\right)$

29.

amplitude: 2; period: 2; midline $y=1;$ equation: $f\left(x\right)=2\cos \left(\pi x\right)+1$

37.

$f\left(x\right)=\text{sin}x$ is symmetric

41.

Maximum: $1$ at $x=0$ ; minimum: $-1$ at $x=\mathrm{\pi }$

43.

A linear function is added to a periodic sine function. The graph does not have an amplitude because as the linear function increases without bound the combined function $h\left(x\right)=x+\text{sin}x$ will increase without bound as well. The graph is bounded between the graphs of $y=x+1$ and $y=x-1$ because sine oscillates between −1 and 1.

45.

There is no amplitude because the function is not bounded.

47.

The graph is symmetric with respect to the y-axis and there is no amplitude because the function's bounds decrease as $\left|x\right|$ grows. There appears to be a horizontal asymptote at $y=0$ .

8.2 Section Exercises

1.

Since $y=\csc x$ is the reciprocal function of $y=\sin x,$ you can plot the reciprocal of the coordinates on the graph of $y=\sin x$ to obtain the y-coordinates of $y=\csc x.$ The x-intercepts of the graph $y=\sin x$ are the vertical asymptotes for the graph of $y=\csc x.$

3.

Answers will vary. Using the unit circle, one can show that $\tan \left(x+\pi \right)=\tan x.$

5.

The period is the same: $2\pi .$

11.

period: 8; horizontal shift: 1 unit to left

17.

$-\cot x\cos x-\sin x$

19.

stretching factor: 2; period: $\frac{\pi }{4};$ asymptotes: $x=\frac{1}{4}\left(\frac{\pi }{2}+\pi k\right)+8,\text{ where}k\text{ is an integer}$

21.

stretching factor: 6; period: 6; asymptotes: $x=3k,\text{ where}k\text{ is an integer}$

23.

stretching factor: 1; period: $\pi ;$ asymptotes: $x=\pi k,\text{ where}k\text{ is an integer}$

25.

Stretching factor: 1; period: $\pi ;$ asymptotes: $x=\frac{\pi }{4}+\pi k,\text{ where}k\text{ is an integer}$

27.

stretching factor: 2; period: $2\pi ;$ asymptotes: $x=\pi k,\text{ where}k\text{ is an integer}$

29.

stretching factor: 4; period: $\frac{2\pi }{3};$ asymptotes: $x=\frac{\pi }{6}k,\text{ where}k\text{ is an odd integer}$

31.

stretching factor: 7; period: $\frac{2\pi }{5};$ asymptotes: $x=\frac{\pi }{10}k,\text{ where}k\text{ is an odd integer}$

33.

stretching factor: 2; period: $2\pi ;$ asymptotes: $x=-\frac{\pi }{4}+\pi k,\text{ where}k\text{ is an integer}$

35.

stretching factor: $\frac{7}{5};$ period: $2\pi ;$ asymptotes: $x=\frac{\pi }{4}+\pi k,\text{ where}k\text{ is an integer}$

37.

$y=\tan \left(3\left(x-\frac{\pi }{4}\right)\right)+2$

39.

$f\left(x\right)=\csc \left(2x\right)$

41.

$f\left(x\right)=\csc \left(4x\right)$

43.

$f\left(x\right)=2\csc x$

45.

$f(x)=\frac{1}{2}\tan (100\pi x)$

55.

1. ⓐ $\left(-\frac{\pi }{2},\frac{\pi }{2}\right);$
2. ⓑ
3. ⓒ $x=-\frac{\pi }{2}$ and $x=\frac{\pi }{2};$ the distance grows without bound as $\left|x\right|$ approaches $\frac{\pi }{2}$ —i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
4. ⓓ3; when $x=-\frac{\pi }{3},$ the boat is 3 km away;
5. ⓔ 1.73; when $x=\frac{\pi }{6},$ the boat is about 1.73 km away;
6. ⓕ 1.5 km; when $x=0$

57.

1. ⓐ $h\left(x\right)=2\tan \left(\frac{\pi }{120}x\right);$
2. ⓑ
3. ⓒ $h\left(0\right)=0:$ after 0 seconds, the rocket is 0 mi above the ground; $h\left(30\right)=2:$ after 30 seconds, the rockets is 2 mi high;
4. ⓓAs $x$ approaches 60 seconds, the values of $h\left(x\right)$ grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

8.3 Section Exercises

1.

The function $y=\sin x$ is one-to-one on $\left[-\frac{\pi }{2},\frac{\pi }{2}\right];$ thus, this interval is the range of the inverse function of $y=\sin x,$ $f(x)={\sin }^{-1}x.$ The function $y=\cos x$ is one-to-one on $\left[0,\pi \right];$ thus, this interval is the range of the inverse function of $y=\cos x,f(x)={\cos }^{-1}x.$

3.

$\frac{\pi }{6}$ is the radian measure of an angle between $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ whose sine is 0.5.

5.

In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ so that it is one-to-one and possesses an inverse.

7.

True . The angle, ${\theta }_1$ that equals $\arccos (-x)$ , $x>0$ , will be a second quadrant angle with reference angle, ${\theta }_2$ , where ${\theta }_2$ equals $\arccos x$ , $x>0$ . Since ${\theta }_2$ is the reference angle for ${\theta }_1$ , ${\theta }_2=\pi -{\theta }_1$ and $\arccos (-x)$ = $\pi -\arccos x$ -

37.

$\frac{x-1}{\sqrt{-x^2+2x}}$

41.

$\frac{x+0.5}{\sqrt{-x^2-x+\frac{3}{4}}}$

49.

domain $\left[-1,1\right];$ range $\left[0,\pi \right]$

51.

approximately $x=0.00$

61.

No. The angle the ladder makes with the horizontal is 60 degrees.

Review Exercises

1.

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

3.

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

5.

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

7.

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

9.

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

11.

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

13.

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

15.

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

17.

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

19.

largest: 20,000; smallest: 4,000

21.

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

23.

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

39.

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

41.

The graphs appear to be identical.

Practice Test

1.

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

3.

amplitude: 5; period: $2\pi ;$ midline: $y=0$

5.

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

7.

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

9.

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

11.

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

13.

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

15.

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

17.

amplitude: 1; period: 12; phase shift: $\mathrm{-6};$ midline $y=\mathrm{-3}$

19.

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

21.

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

23.

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

27.

The views are different because the period of the wave is $\frac{1}{25}.$ Over a bigger domain, there will be more cycles of the graph.

31.

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

33.

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

35.

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

41.

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

49.

approximately 0.07 radians

College Algebra & Trigonometry 5th Edition Chapter 8 Practice Questions

Source: https://openstax.org/books/algebra-and-trigonometry/pages/chapter-8

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