AP Pre-Calc Unit 3 Cheatsheet: Trigonometric and Polar Functions

Periodic relationship: Output values repeat in a predictable pattern as input increases over successive equal-length intervals.

• Period k: smallest positive value where f(x + k) = f(x) for all x in domain
• Behavior in one period completely determines the entire function
• All characteristics (increase/decrease, concavity, rate of change) repeat in every period

Estimating period: Find where successive equal-length outputs begin repeating; investigate the graph or data table for the smallest repeating cycle.

Standard position: Vertex at origin, one ray on positive x-axis (initial ray), other ray is terminal ray.

• Positive angles: counterclockwise rotation
• Negative angles: clockwise rotation
• Coterminal angles: differ by integer multiples of 2π (or 360°)

Radian measure: θ (in radians) = arc length ÷ radius. On unit circle (r = 1), radian measure equals arc length.

Key conversions: π rad = 180°; 2π rad = 360°

For angle θ in standard position, point P = (cos θ, sin θ) on unit circle.

• sin θ = y-coordinate of P = vertical displacement
• cos θ = x-coordinate of P = horizontal displacement
• Both range: [−1, 1]
• Both have period 2π

For circle radius r: Point P = (r cos θ, r sin θ)

Exact values: Use isosceles right triangles (π/4, π/4, π/2) and 30°–60°–90° triangles formed by bisecting an equilateral triangle (π/6, π/3, π/2), with attention to quadrant signs.

Definition: tan θ = sin θ / cos θ (where cos θ ≠ 0) = slope of terminal ray

• Period: π (repeats every half-revolution)
• Vertical asymptotes: θ = π/2 + kπ (k integer), where cos θ = 0
• Increases and changes concavity (down to up) between consecutive asymptotes
• Domain: all reals except π/2 + kπ
• Range: all real numbers

Transformations: y = a tan(b(θ + c)) + d

• Vertical scale factor: |a| (tangent has no amplitude)
• Period: |1/b| · π
• Phase shift: −c units
• Vertical shift: d units