AP Precalculus Unit 3 Cheat Sheet: Trigonometric & Polar Functions

The basics

What it covers: Periodic behavior — trigonometric functions from the unit circle, sinusoidal modeling, tangent and reciprocal functions, inverse trig, trig equations, and polar functions.

Exam weight: About 30–35% of the AP Precalculus exam — and it is the longest unit.

The big question: How do the trigonometric functions, built from the unit circle, model repeating phenomena, and how does the polar system give a second way to graph them?

Mathematical practices: Procedural & Symbolic Fluency (P1), Multiple Representations (P2), Communication & Reasoning (P3).

Key topics at a glance

Periodic Basics

Period = length of one cycle. Amplitude = half the max−min distance = |a|. Midline = the value the graph oscillates around = d.

The Unit Circle

For angle θ, the point is (cos θ, sin θ). tan θ = sin θ / cos θ. Radians: a full circle = 2π. Signs by quadrant: All, Sine, Tangent, Cosine positive (I–IV).

Sine & Cosine Graphs

Domain all reals, range [−1, 1], period 2π. Cosine starts at a maximum; sine starts at the midline going up.

Sinusoidal Form

f(x) = a·sin(b(x − c)) + d: amplitude |a|, period = 2π/|b|, phase shift c, midline d.

Tangent & Reciprocals

Tangent: period π, asymptotes where cos = 0. sec = 1/cos, csc = 1/sin, cot = cos/sin; asymptotes where the denominator is 0.

Inverse Trig & Equations

Restrict domains so trig is one-to-one: arcsin/arctan output [−π/2, π/2], arccos outputs [0, π]. Trig equations have infinitely many solutions (add multiples of the period).

Identities

Pythagorean: sin²θ + cos²θ = 1, and 1 + tan²θ = sec²θ. Even/odd: cos is even; sin and tan are odd.

Polar Functions

Point (r, θ); convert with x = r cos θ, y = r sin θ. Graphs: circle, rose, limaçon, cardioid, spiral. Rate of change = Δr/Δθ.

The key terms you must know

Period / amplitude / midline — length of one cycle; half the max−min distance; the center line y = d.
Radian — angle whose arc length equals the radius; a full circle is 2π radians.
Unit circle — for angle θ, the point (cos θ, sin θ); tangent is sin θ / cos θ.
Sinusoidal form — f(x) = a·sin(b(x − c)) + d, with amplitude |a|, period 2π/|b|, phase shift c, midline d.
Phase shift — the horizontal shift c of a sinusoidal graph.
Tangent function — period π; asymptotes where cos = 0; increasing between asymptotes.
Reciprocal functions — sec = 1/cos, csc = 1/sin, cot = cos/sin.
Inverse trig functions — arcsin, arccos, arctan, defined on restricted domains to be one-to-one.
Pythagorean identity — sin²θ + cos²θ = 1.
Polar coordinates — (r, θ): directed distance r and angle θ from the polar axis.
Polar-to-rectangular — x = r cos θ, y = r sin θ; r = √(x² + y²).

Key themes to remember

Everything starts on the unit circle. Sine and cosine are coordinates; every graph, identity, and value traces back to it.
Sinusoidal functions are transformations. a, b, c, d each map to amplitude, period, phase shift, and midline.
Periodicity means infinitely many solutions. Solve one, then add multiples of the period.
Reciprocals and inverses are different ideas. Secant is 1/cosine; arccosine undoes cosine — do not confuse them.
Polar is a second coordinate system. The same point and curve can be described with (r, θ) or (x, y).

Common exam traps

Reciprocal ≠ inverse. sec x = 1/cos x, but arccos x undoes cosine. sin⁻¹ x means arcsine, not 1/sin x.
Period is 2π/|b|, not 2π·|b|. A larger b gives a shorter period (more cycles).
Phase shift direction. In a·sin(b(x − c)) + d the graph shifts right by c; a plus sign inside shifts left.
Inverse trig has a restricted range. arcsin only returns values from −π/2 to π/2; a trig equation may need additional solutions from periodicity.
Radians vs. degrees. On the AP Exam, work in radians unless told otherwise; check your calculator's mode.
Amplitude is |a|, always positive. A negative a reflects the graph but the amplitude is still |a|.
tan and sec share asymptotes with cos = 0; csc and cot with sin = 0. Match the asymptote to the zero of the denominator.