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 2π 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 θ in radians, the arc length is s=rθ (not valid as written if θ is in degrees).

Degree–radian equivalence

360° = 2π radians, so 180° = π radians.

Degree to radian conversion

θrad = θdeg · (π/180).

Radian to degree conversion

θdeg = θrad · (180/π).

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(θ)=sin(θ)/cos(θ) (when cos(θ)≠0); on the unit circle tan(θ)=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 θ, −1 ≤ sin(θ) ≤ 1.

Cosine function range

For all real θ, −1 ≤ cos(θ) ≤ 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(θ) is undefined where cos(θ)=0, i.e., at θ = (π/2) + kπ for any integer k.

Period of tangent

tan(θ+π)=tan(θ); tangent repeats every π (not 2π).

Even/odd trig symmetry

cos is even: cos(−θ)=cos(θ); sin is odd: sin(−θ)=−sin(θ); tan is odd: tan(−θ)=−tan(θ).

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=2π/|B|, C is horizontal (phase) shift, and D is vertical shift/midline (y=D).