Trigonometric Functions - I — Formula Sheet

For any point P(x, y) on the unit circle corresponding to angle θ, we define: sin θ = y, cos θ = x.

tan θ = y/x (defined when x ≠ 0), cot θ = x/y (defined when y ≠ 0).

Memory trick for sin values

sin 0 = √(0/4), sin π/6 = √(1/4), sin π/4 = √(2/4), sin π/3 = √(3/4), sin π/2 = √(4/4). Simplify to get 0, 1/2, 1/√2, √3/2, 1.

Cosine values are the reverse

cos 0 = 1, cos π/6 = √3/2, cos π/4 = 1/√2, cos π/3 = 1/2, cos π/2 = 0.

Graph of y = sin θ: starts at 0, rises to maximum 1 at π/2, returns to 0 at π, falls to minimum -1 at 3π/2, returns to 0 at 2π. It is continuous and wave-shaped.

Graph of y = cos θ: starts at maximum 1 at θ=0, falls to 0 at π/2, reaches minimum -1 at π, rises to 0 at 3π/2, returns to 1 at 2π. It is also continuous and wave-shaped.

Graph of y = tan θ: increases from 0 to +∞ in (0, π/2), then from -∞ to 0 in (π/2, π). Has vertical asymptotes at θ = π/2 and 3π/2. Discontinuous.

A unit circle (radius = 1) centered at the origin provides the geometric foundation for defining trigonometric functions.

Since (x, y) coordinates on the unit circle satisfy x² + y² = 1, we have -1 ≤ cos θ ≤ 1 and -1 ≤ sin θ ≤ 1.