Trigonometric Identities & Equations Solver

Trigonometric Identities Guide

Pythagorean Identities

sin²A + cos²A = 1 (the fundamental Pythagorean identity). Derived: tan²A + 1 = sec²A. 1 + cot²A = cosec²A. Example: if sin A = 3/5, find cos A. cos²A = 1 − 9/25 = 16/25. cos A = ±4/5 (sign depends on quadrant). These identities convert between sin, cos, and tan in simplification or solving equations. Combined with the unit circle, they allow finding all trig values from any one given value (within ambiguity over quadrant). Most-used identity in calculus and physics.

Double Angle Identities

sin(2A) = 2 sin A cos A. cos(2A) = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1. tan(2A) = 2 tan A / (1 − tan²A). Example: sin(2 × 30°) = sin(60°) = √3/2 ≈ 0.866. Verify: 2 × sin(30°) × cos(30°) = 2 × 0.5 × (√3/2) = √3/2 ✓. Applications: integration (∫sin²x dx uses cos(2x) = 1 − 2sin²x → sin²x = (1−cos(2x))/2). Wave physics (intensity proportional to amplitude², leading to factor-of-2 doubling). Used in solving equations like sin(2x) = sin(x): 2 sin x cos x = sin x → sin x (2 cos x − 1) = 0 → x = 0°,

Sum and Difference Identities

sin(A + B) = sin A cos B + cos A sin B. sin(A − B) = sin A cos B − cos A sin B. cos(A + B) = cos A cos B − sin A sin B. cos(A − B) = cos A cos B + sin A sin B. tan(A + B) = (tan A + tan B) / (1 − tan A tan B). Useful for: exact values (sin 75° = sin(45° + 30°) using known exact values for 30° and 45°). Reducing complicated expressions to known angles. Phasor addition in physics — adding two oscillations with different phases. AC circuit analysis uses these identities constantly.

Half-Angle Identities

sin(A/2) = ±√((1 − cos A)/2). cos(A/2) = ±√((1 + cos A)/2). tan(A/2) = (1 − cos A)/sin A = sin A/(1 + cos A). Sign depends on which quadrant A/2 is in. Weierstrass substitution: t = tan(A/2). sin A = 2t/(1+t²). cos A = (1−t²)/(1+t²). Converts any rational function of trig to a rational function of t — making many integrals possible to solve analytically. Half-angle identities also useful in solving complex trig equations and in the derivation of geometric formulas for triangle areas.