AP Precalculus Unit 3 Trigonometric Functions: Understanding Cycles, Graphs, and Models

Periodic phenomenon

A behavior or process that repeats in a predictable cycle; after a fixed input change, the output pattern repeats.

Period (P)

The fixed repeat length of a periodic function; a number P>0 such that f(x+P)=f(x).

Periodic function (definition)

A function f is periodic if there exists P>0 such that f(x+P)=f(x) for all x in the domain.

Fundamental period

The smallest positive period P (if it exists) for which f(x+P)=f(x).

Degrees

An angle unit where one full rotation equals 360°.

Radians

An angle unit where one full rotation equals $2\pi$ radians; the natural unit for trigonometry because it ties directly to arc length.

Arc length formula (radians only)

For a circle of radius $r$ and central angle $\theta$ in radians, the arc length is $s=r\theta$ (not valid as written if $\theta$ is in degrees).

Degree–radian equivalence

360° = $2\pi$ radians, so 180° = $\pi$ radians.

Degree to radian conversion

$\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$.

Radian to degree conversion

$\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}$.

Unit circle

The circle of radius 1 centered at the origin; equation $x^2 + y^2 = 1$.

Sine (unit circle definition)

For angle θ, sin(θ) is the y-coordinate of the point on the unit circle reached by rotating θ from (1,0).

Cosine (unit circle definition)

For angle θ, cos(θ) is the x-coordinate of the point on the unit circle reached by rotating θ from (1,0).

Tangent (definition)

tan($\theta$) = $\frac{\sin(\theta)}{\cos(\theta)}$ (when $\cos(\theta) \neq 0$); on the unit circle tan($\theta$) = $\frac{y}{x}$ and corresponds to the slope of the terminal ray (except vertical rays).

Reference angle

The acute angle between the terminal side of θ and the x-axis; used to find trig values via symmetry and quadrant signs.

Quadrant sign rule (unit circle)

Use the same reference-angle magnitudes as in Quadrant I, but assign signs to x (cos) and y (sin) based on the quadrant.

Special angles (Quadrant I)

Common angles used for exact trig values: 0, π/6, π/4, π/3, π/2 (from 30-60-90 and 45-45-90 triangles).

Sine function range

For all real $\theta$, $-1 \le \sin(\theta) \le 1.$

Cosine function range

For all real $\theta$, $-1 \le \cos(\theta) \le 1$.

Period of sine and cosine

Both repeat every 2π: sin(θ+2π)=sin(θ) and cos(θ+2π)=cos(θ).

Tangent domain restriction

$\tan(\theta)$ is undefined where $\cos(\theta)=0$, i.e., at $\theta = \frac{\pi}{2} + k\pi$ for any integer $k$.

Period of tangent

tan($\theta + \pi$) = tan($\theta$); tangent repeats every $\pi$ (not $2\pi$).

Even/odd trig symmetry

$\cos$ is even: $\cos(-\theta)=\cos(\theta)$; $\sin$ is odd: $\sin(-\theta)=-\sin(\theta)$; $\tan$ is odd: $\tan(-\theta)=-\tan(\theta)$.

Amplitude

The size of vertical variation in a sinusoid; for y=A \sin(…) + D or y=A \cos(…) + D, amplitude is |A|.

General sinusoidal model (parameter meanings)

y=A \sin(B(x−C))+D or y=A \cos(B(x−C))+D where |A| is amplitude, period P=\frac{2\pi}{|B|}, C is horizontal (phase) shift, and D is vertical shift/midline (y=D).