Using Multiple Angle Formulas

Pre-Calculus · Axiom Academy

EXAMPLE Using Multiple-Angle Formulas Given one ratio and a quadrant, build the double-angle values step by step. Suppose and x lies in Quadrant II . Find , , and . In Quadrant II, sine is positive and cosine is negative — that sign decides the root we keep. Nice work. You turned a single ratio plus a quadrant into the full set of double-angle values. Sign first: the Pythagorean identity gives the missing ratio's size ; the quadrant fixes its sign . In Q2, . Substitute, don't memorize endings: and , then . Half-angle, same idea: read off a half-angle by halving the inside angle — e.g. , taking + because is in Q1. Run formulas in reverse to simplify: spotting the pattern collapses straight to . Every multiple-angle problem is the same loop: find the missing ratio (sign from the quadrant), drop it into the formula, simplify.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.