AP Precalculus — Trigonometric and Polar Functions flashcards

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Unit 3 of AP Precalculus: periodic behavior, the unit circle, sinusoidal functions and models, tangent and inverse trig functions, equations and inequalities, reciprocal functions, identities, and polar coordinates and graphs.

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Learn deck · part of AP Precalculus · written against: College Board AP Precalculus Course and Exam Description (Units 1–3 (exam-assessed units)). Updated 2026-10-10.

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What makes a function periodic, and what is its period?

$f$ is periodic if $f(x + P) = f(x)$ for every $x$ and some positive constant $P$ ⁠; the period is the smallest such $P$ ⁠, the input length of one full cycle.
Trigonometric and Polar Functions › Periodic Phenomena
A rider's height on a Ferris wheel is graphed against time. What do the period, amplitude and midline represent?

Period = time for one full rotation; amplitude = the wheel's radius; midline = the height of the wheel's center.
Trigonometric and Polar Functions › Periodic Phenomena
Geometrically, what does an angle of 1 radian measure?

A central angle whose intercepted arc is as long as the circle's radius, so a full turn is $2\pi$ radians.
Trigonometric and Polar Functions › Periodic Phenomena
$f$ has period 6 and $f(2) = 5$ ⁠. What is $f(20)$ ⁠?

5, because $20 = 2 + 3 \cdot 6$ and shifting the input by any whole number of periods leaves the output unchanged.
Trigonometric and Polar Functions › Periodic Phenomena
How are $\sin \theta$ and $\cos \theta$ defined from the unit circle?

The terminal ray of $\theta$ in standard position meets the unit circle at the point $(\cos \theta, \sin \theta)$ ⁠: cosine is the x-coordinate and sine the y-coordinate.
Trigonometric and Polar Functions › The Unit Circle
How is $\tan \theta$ related to the terminal ray of $\theta$ ⁠?

$\tan \theta = \frac{\sin \theta}{\cos \theta}$ is the slope of the terminal ray, so it is undefined whenever the ray is vertical.
Trigonometric and Polar Functions › The Unit Circle
The terminal ray of $\theta$ passes through $(-3, 4)$ ⁠. What are $\sin \theta$ ⁠, $\cos \theta$ and $\tan \theta$ ⁠?

The distance from the origin is $r = 5$ ⁠, so $\sin \theta = \frac{4}{5}$ ⁠, $\cos \theta = -\frac{3}{5}$ and $\tan \theta = -\frac{4}{3}$ ⁠.
Trigonometric and Polar Functions › The Unit Circle
Where does a point at angle $\theta$ lie on a circle of radius $r$ centered at the origin?

At $(r\cos \theta, r\sin \theta)$ ⁠.
Trigonometric and Polar Functions › The Unit Circle
In which quadrants are sine, cosine and tangent positive?

Quadrant I: all three; II: sine; III: tangent; IV: cosine.

Mnemonic: "All Students Take Calculus," going counterclockwise from Quadrant I.
Trigonometric and Polar Functions › The Unit Circle
What are the exact sine and cosine values at $\frac{\pi}{6}$ ⁠, $\frac{\pi}{4}$ and $\frac{\pi}{3}$ ⁠?

$\sin\frac{\pi}{6} = \cos\frac{\pi}{3} = \frac{1}{2}$ ⁠, $\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ ⁠, $\sin\frac{\pi}{3} = \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$ ⁠.
Trigonometric and Polar Functions › The Unit Circle

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Review status: Second-reader reviewed, with corrections applied.

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Topic outline follows the College Board AP Precalculus Course and Exam Description (effective Fall 2026), Unit 3 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Precalculus 2e and OpenStax Calculus Volumes 1 and 2, consulted as references; no text copied.

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