Question 1

A ladder leans against a wall, forming a right triangle with the ground. If the ladder is 10 feet long and the angle between the ladder and the ground is $$30^\circ$$, what is the height at which the ladder touches the wall?

Accepted Answer

$$10 \sin(30^\circ) = 5$$ feet

Question 2

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

Accepted Answer

$$\dfrac{3\pi}{4}$$

Question 3

Given $$\sin(	heta) = \dfrac{3}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos(	heta)$$?

Accepted Answer

$$-\dfrac{4}{5}$$

Question 4

A central angle of $$2$$ radians in a circle of radius $$6$$ meters subtends an arc. What is the length of the arc?

Accepted Answer

$$12$$ meters

Question 5

Which of the following is the correct equation for the unit circle?

Accepted Answer

$$x^2$$ + $$y^2 = 1$

Question 6

If $$\sin(\alpha) = \dfrac{5}{13}$$ and $$\alpha is in $$Quadrant I, what is $$\cos(\alpha)$$?

Accepted Answer

$$\dfrac{12}{13}$$

Question 7

Evaluate $$\sin(75^\circ)$$ using the sum formula.

Accepted Answer

$$\sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)$$

Question 8

A Ferris wheel has a radius of $$20$$ meters and makes one complete revolution every $$40$$ seconds. If a passenger starts at the lowest point, what is the height above the ground after $$10$$ seconds? (Assume the center is $$22$$ meters above the ground.)

Accepted Answer

$$22 + 20\sin\left(\dfrac{\pi}{2}\right) = 42$$ meters

Question 9

Which of the following is the correct Pythagorean identity?

Accepted Answer

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

Question 10

If $$	an(	heta) = 3$$ and $$	heta is in $$Quadrant III, what is $$\sin(	heta)$$?

Accepted Answer

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

Question 11

Find the exact value of $$\cos(2x) if$$ $$\sin(x) = \dfrac{3}{5}$$ and $$x is in $$Quadrant II.

Accepted Answer

$$-\dfrac{7}{25}$$

Question 12

Which transformation has been applied to $$y = \sin(x) to $$obtain $$y = 2\sin(x - \dfrac{\pi}{4})$$?

Accepted Answer

Vertical stretch by 2 and horizontal shift right by $$\dfrac{\pi}{4}$$

Question 13

Solve for $$x in$$ $$2\sin(x) - 1 = 0$$ for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

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

Question 14

If $$\sin(	heta) = -\dfrac{1}{2}$$ and $$	heta is in$$ $$[0, 2\pi)$$, what are all possible values of $$	heta$$?

Accepted Answer

$$\dfrac{7\pi}{6}, \dfrac{11\pi}{6}$$

Question 15

Which of the following is the correct sum-to-product formula for $$\sin(A) + \sin(B)$$?

Accepted Answer

$$2\sin\left(\dfrac{A+B}{2}\right)\cos\left(\dfrac{A-B}{2}\right)$$

Question 16

A sound wave can be modeled by $$y = 3\sin(2\pi t - \dfrac{\pi}{3}). $$What is the period of the wave?

Accepted Answer

$$1$$ second

Question 17

If $$	an(\alpha) = 2$$ and $$\alpha is in $$Quadrant I, what is $$\sin(2\alpha)$$?

Accepted Answer

$$\dfrac{4}{5}$$

Question 18

Which of the following is the correct reciprocal identity?

Accepted Answer

$$\csc(x) = \dfrac{1}{\sin(x)}$$

Question 19

Given $$\cos(	heta) = -\dfrac{3}{5}$$ and $$	heta is in $$Quadrant III, what is $$	an(	heta)$$?

Accepted Answer

$$\dfrac{4}{3}$$

Question 20

If $$\sin(x) = \dfrac{1}{2}$$ and $$x is in$$ $$[0, 2\pi)$$, what are all possible values of $$x$$?

Accepted Answer

$$\dfrac{\pi}{6}, \dfrac{5\pi}{6}$$

Comprehensive Study Notes: Analytic and Applied Trigonometry for Precalculus

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