Question 1

Which of the following best describes the unit circle and its significance in trigonometry?

Accepted Answer

The unit circle is a circle with radius 1 centered at the origin, and it allows us to define trigonometric functions for all real numbers using the coordinates of points on the circle.

Question 2

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

Accepted Answer

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

Question 3

A Ferris wheel has a radius of 20 meters and completes one revolution every 2 minutes. If a rider starts at the lowest point, what is the rider's height above the center after $$\dfrac{1}{4} of a $$revolution?

Accepted Answer

$$20$$ meters

Question 4

Which of the following is the correct graph for $$y = 2\sin(x) + 1$$?

Accepted Answer

A sine wave with amplitude 2, midline at $$y=1$$, and period $$2\pi.$$

Question 5

Solve for $$x in$$ $$\sin(x) = \dfrac{1}{2}$$, where $$0 \leq x \u003c 2\pi.$$

Accepted Answer

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

Question 6

Which of the following is the correct expression for $$\cos(2	heta) in $$terms of $$\sin(	heta)$$?

Accepted Answer

$$1 - 2\sin^2$(	heta)$$

Question 7

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

Accepted Answer

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

Question 8

Which of the following is the correct formula for the Law of Sines?

Accepted Answer

$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$$

Question 9

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 120^\circ. $$Using the Law of Cosines, what is the length of side $$c$$?

Accepted Answer

$$c = \sqrt{7^2$ + 10^2$ - 2 \cdot 7 \cdot 10 \cdot \cos(120^\circ)}$$

Question 10

Which of the following is the correct inverse function for $$y = \sin(x)$$, where $$-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}$$?

Accepted Answer

$$x = \arcsin(y)$$

Question 11

If $$z = 4	ext{cis}\left(\dfrac{\pi}{3}ight)$$, what is the rectangular form of $$z$$?

Accepted Answer

$$z = 2 + 2\sqrt{3}i$$

Question 12

Which of the following points is equivalent to $$(3, \dfrac{\pi}{2}) in $$polar coordinates?

Accepted Answer

$$(0, 3) in$\ (x, y)$$\ coordinates

Question 13

Expand $$(x + 2)^4$ using the Binomial Theorem.

Accepted Answer

$$x^4$$ + $$8x^3$$ + $$24x^2 + 32x + 16$

Question 14

A sound wave can be modeled by $$y = 0.5\cos(2\pi t - \dfrac{\pi}{3}). $$What is the phase shift of this wave?

Accepted Answer

$$\dfrac{1}{6}$$ units to the right

Question 15

Which of the following is an identity?

Accepted Answer

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

Question 16

Solve for $$x$$: $$2\cos^2$(x) - 1 = 0$$, where $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = 0,\ \pi$$

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$

Question 19

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

Accepted Answer

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

Precalculus Trigonometry: Course Syllabus and Study Guide

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