Trigonometric Functions — Formula Sheet

One complete revolution = 360° = 2π radians.

A unit circle has centre at origin O and radius = 1. For any point P(a, b) on the unit circle with angle AOP = x radians (measured from positive x-axis), we DEFINE: cos x = a (x-coordinate) and sin x

Since P lies on the unit circle

a² + b² = 1, giving the fundamental identity: sin²x + cos²x = 1.

The other four functions are defined as

tan x = sin x / cos x, cot x = cos x / sin x, sec x = 1 / cos x, cosec x = 1 / sin x.

Core Sum Formulas

sin(x+y) = sin x cos y + cos x sin y. cos(x+y) = cos x cos y – sin x sin y. tan(x+y) = (tan x + tan y)/(1 – tan x tan y).

Core Difference Formulas

sin(x–y) = sin x cos y – cos x sin y. cos(x–y) = cos x cos y + sin x sin y. tan(x–y) = (tan x – tan y)/(1 + tan x tan y).

For y = sin x and y = cos x: Domain is all real numbers ℝ; Range is [-1, 1].