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AP Precalculus: Trigonometric and Polar Functions — Practice Questions & Answers
Unit 3 extends function thinking to periodic behavior. You will measure angles in radians on the unit circle, build and transform the graphs of sine, cosine, and tangent, model real cyclical data with sinusoids, invert trig functions, solve trig equations, apply core identities, and finally describe curves in the polar coordinate system.
170 practice questions available for this topic — here are 10 with full answers and explanations.
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Practice questions with answers
1
Multiple choice · Easy
Convert $180^{\circ}$ to radians.
$\pi$
$2\pi$
$\frac{\pi}{2}$
$\frac{\pi}{180}$
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Why Multiply degrees by $\frac{\pi}{180}$, so $180\cdot\frac{\pi}{180}=\pi$.
2
Multiple choice · Easy
What is the amplitude of $y=3\sin(x)$?
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Why The amplitude is the absolute value of the coefficient of the sine, which is $|3|=3$.
3
Multiple choice · Easy
What is the period of $y=\sin(x)$?
$\pi$
$\frac{\pi}{2}$
$2\pi$
$1$
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Why The basic sine function completes one full cycle over an interval of length $2\pi$.
4
Multiple choice · Easy
Evaluate $\sin\left(\frac{\pi}{6}\right)$.
$\frac{\sqrt{3}}{2}$
$\frac{\sqrt{2}}{2}$
$\frac{1}{2}$
$1$
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Why On the unit circle, $\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$.
5
Multiple choice · Easy
What is the midline of $y=\sin(x)+4$?
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Why The vertical shift of $+4$ moves the midline to $y=4$.
6
Multiple choice · Easy
Evaluate $\cos(0)$.
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Why At an angle of $0$, the terminal point on the unit circle is $(1,0)$, so $\cos(0)=1$.
7
Multiple choice · Easy
What is the period of $y=\cos(2x)$?
$2\pi$
$\pi$
$4\pi$
$\frac{\pi}{2}$
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Why The period is $\frac{2\pi}{b}=\frac{2\pi}{2}=\pi$.
8
Fill in the blank · Easy
Convert $90^{\circ}$ to radians. Express the result in terms of $\pi$:
Check answer
Answer:
pi/2 / \frac{\pi}{2} / 1.5708
Why Multiply by $\frac{\pi}{180}$: $90\cdot\frac{\pi}{180}=\frac{\pi}{2}$.
9
Fill in the blank · Easy
The amplitude of $y=-5\cos(x)$ is
Check answer
Answer:
5
Why Amplitude is the absolute value of the coefficient, $|-5|=5$.
10
Multiple choice · Medium
What is the phase shift of $y=\sin\left(x-\frac{\pi}{3}\right)$?
Right by $\frac{\pi}{3}$
Left by $\frac{\pi}{3}$
Right by $\frac{\pi}{6}$
Up by $\frac{\pi}{3}$
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Why A subtraction inside the function shifts the graph to the right by $\frac{\pi}{3}$.
Key terms in Trigonometric and Polar Functions
Periodic Function: A function whose values repeat at regular intervals of the input.
Period: The horizontal length of one complete cycle of a periodic function.
Cycle: One complete repetition of a periodic pattern.
Amplitude: Half the difference between the maximum and minimum values of a periodic function.
Midline: The horizontal line halfway between the maximum and minimum of a periodic function.
Frequency: The number of cycles a periodic function completes per unit of input.
Phase Shift: A horizontal translation of a periodic function.
Vertical Shift of Sinusoid: A translation of a sinusoidal graph up or down changing the midline.
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