Question 1

Which of the following is the correct domain for the function $$f(x) = \arcsin(x)$$?

Accepted Answer

$$[-1, 1]$$

Question 2

A Ferris wheel with a radius of 20 meters completes one revolution every 40 seconds. If a rider starts at the lowest point, which of the following sinusoidal functions best models the rider's height $$h(t)$$ above the ground (in meters) as a function of time $$t (in $$seconds), assuming the center of the wheel is 22 meters above the ground?

Accepted Answer

$$h(t) = 20 \cos\left(\frac{\pi}{20} t\right) + 22$$

Question 3

Solve the equation $$2\sin(x) - 1 = 0$$ for all solutions in the interval $$[0, 2\pi).$$

Accepted Answer

$$x = \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 4

Which of the following expressions is equivalent to $$\sin(\arccos(x))$$ for $$-1 \leq x \leq 1$$?

Accepted Answer

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

Question 5

Given $$f(x) = 3\sin(2x - \frac{\pi}{3}) + 1$$, what is the amplitude of the function?

Accepted Answer

$$3$$

Question 6

If $$	an(	heta) = 3$$ and $$	heta is in $$the second quadrant, what is $$\sin(	heta)$$?

Accepted Answer

$$-\frac{3}{\sqrt{10}}$$

Question 7

Which of the following is the range of the function $$f(x) = \arctan(x)$$?

Accepted Answer

$$(-\frac{\pi}{2}, \frac{\pi}{2})$$

Question 8

A mass on a spring oscillates according to $$y(t) = 5\cos(\pi t)$$, where $$y is $$the displacement in centimeters and $$t is $$time in seconds. What is the period of the motion?

Accepted Answer

$$2$$ seconds

Question 9

Which of the following is the correct inverse function of $$f(x) = 	an(x)$$, where $$x is $$restricted to $$\left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$$?

Accepted Answer

$$f^{-1}(x$$) = \arctan(x)$$

Question 10

If $$\sin(	heta) = \frac{4}{5}$$ and $$	heta is in $$the first quadrant, what is $$\cos(	heta)$$?

Accepted Answer

$$\frac{3}{5}$$

Question 11

Which of the following best describes the effect of the coefficient $$b in $$the function $$y = a\sin(bx + c) + d$$?

Accepted Answer

It determines the period of the graph.

Question 12

Solve for $$x$$: $$\cos(2x) = 0$$ for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Question 13

Which of the following is the correct composition: $$\sin(\arccos(x))$$?

Accepted Answer

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

Question 14

A sound wave can be modeled by $$y(t) = 0.5\sin(440\pi t)$$, where $$t is in $$seconds. What is the frequency of the wave in Hz?

Accepted Answer

$$220 Hz$$

Question 15

Which of the following is the range of the function $$f(x) = \arccos(x)$$?

Accepted Answer

$$[0, \pi]$$

Question 16

If $$\sin(x) = 0.6$$, what is $$x in $$radians, rounded to two decimal places, for $$x in$$ $$[0, \pi]$$?

Accepted Answer

$$x \approx 0.64 or$$ $$x \approx 2.50$$

Question 17

Which of the following is the period of the function $$y = 2\sin(3x)$$?

Accepted Answer

$$\frac{2\pi}{3}$$

Question 18

If $$f(x) = \sin(x)$$ and $$g(x) = \arcsin(x)$$, what is $$f(g(0.5))$$?

Accepted Answer

$$0.5$$

Question 19

A daylight model for a city is $$D(t) = 3\sin\left(\frac{2\pi}{365} t - 1.4\right) + 12$$, where $$D is $$the number of daylight hours and $$t is $$the day of the year. What is the average number of daylight hours throughout the year?

Accepted Answer

$$12$$

Precalculus Study Notes: Trigonometric Functions, Graphs, and Applications

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