Q: A circular track has a radius of 50 meters. How far does a runner travel if she runs through a central angle of $$72^\circ$$?

$$50 \times \frac{72\pi}{180} = 20\pi$$ meters

Q: Which of the following is the correct identity for $$\tan(\theta) in $$terms of $$\sin(\theta)$$ and $$\cos(\theta)$$?

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

Q: A building casts a shadow 30 meters long when the angle of elevation of the sun is $$40^\circ. $$How tall is the building?

$$30 \tan 40^\circ \approx 25.2$$ meters

Q: Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$\theta (in $$radians)?

$$A = \frac{1}{2} r^2$ \theta$$

Question 1

A Ferris wheel has a radius of 20 meters and completes one full revolution every 2 minutes. If a passenger starts at the lowest point, what is the angular speed of the Ferris wheel in radians per second?

Accepted Answer

$$\frac{\pi}{30}$$

Question 2

Convert $$135^\circ to $$radians.

Accepted Answer

$$\frac{3\pi}{4}$$

Question 3

A ladder leans against a wall forming a $$60^\circ$$ angle with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Assume the wall is perpendicular to the ground.)

Accepted Answer

$$10 \sin 60^\circ = 5\sqrt{3} ft$$

Question 4

Which of the following is the correct definition of the sine function for a general angle $$	heta in $$the unit circle?

Accepted Answer

The $$y-$$coordinate of the point where the terminal side of $$	heta$$ intersects the unit circle.

Question 5

If $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos 	heta$$?

Accepted Answer

$$-\frac{4}{5}$$

Question 6

A circular track has a radius of 50 meters. How far does a runner travel if she runs through a central angle of $$72^\circ$$?

Accepted Answer

$$50 	imes \frac{72\pi}{180} = 20\pi$$ meters

Question 7

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

Accepted Answer

$$\pi$$

Question 8

The graph of $$y = \cos(x) is $$shifted $$\frac{\pi}{3}$$ units to the right. What is the equation of the new graph?

Accepted Answer

$$y = \cos(x - \frac{\pi}{3})$$

Question 9

Which of the following is the correct identity for $$	an(	heta) in $$terms of $$\sin(	heta)$$ and $$\cos(	heta)$$?

Accepted Answer

$$	an(	heta) = \frac{\sin(	heta)}{\cos(	heta)}$$

Question 10

If $$\sin(x) = 0.6$$ and $$x is in $$the first quadrant, what is $$\arcsin(0.6) in $$degrees (rounded to the nearest degree)?

Accepted Answer

$$37^\circ$$

Question 11

A building casts a shadow 30 meters long when the angle of elevation of the sun is $$40^\circ. $$How tall is the building?

Accepted Answer

$$30 	an 40^\circ \approx 25.2$$ meters

Question 12

Which of the following is the correct graph for $$y = 	an(x)$$?

Accepted Answer

A curve with vertical asymptotes at $$x = \frac{\pi}{2} + n\pi$$, $$n \in \mathbb{Z}$$

Question 13

If $$\cos(	heta) = -\frac{1}{2}$$ and $$	heta is in $$the third quadrant, what is $$	heta in $$degrees?

Accepted Answer

$$240^\circ$$

Question 14

Which of the following is an identity?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 15

Find the amplitude of the function $$y = -4 \sin(3x).$$

Accepted Answer

$$4$$

Question 16

Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$	heta (in $$radians)?

Accepted Answer

$$A = \frac{1}{2} r^2$ 	heta$$

Question 17

If $$	an(\alpha) = 2$$ and $$\alpha is in $$the first quadrant, what is $$\sin(\alpha)$$?

Accepted Answer

$$\frac{2}{\sqrt{5}}$$

Question 18

Which of the following is the correct inverse function for $$y = \sin(x)$$, where $$x is $$restricted to $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$?

Accepted Answer

$$y = \arcsin(x)$$

Question 19

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

Accepted Answer

$$220 Hz$$

Question 20

Which of the following is the correct expression for $$\sin(2x)$$ using a double-angle identity?

Accepted Answer

$$2 \sin(x) \cos(x)$$

Precalculus Study Notes: Trigonometric Functions and Their Graphs

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