Half-Angle Formulas

Pre-Calculus · Axiom Academy

Cut any angle in half — and read off its sine, cosine, and tangent exactly. The catch is a single ± sign, decided by one thing. 1. Where the Formulas Come From Start from a double-angle identity you already know and just solve it for the half-angle. Take the form , isolate , then set (so ). The animation rearranges it term by term. 2. The Four Half-Angle Formulas The sine and cosine forms keep a because they came through a square root. The two tangent forms do not — they are built as a ratio , so the sign takes care of itself. The animation shows that ratio assembling on the unit circle, no radical in sight. This is the whole game. The is decided by the quadrant of the half-angle — the same rule as any sine or cosine: positive where that function is positive, negative where it is negative. In the animation the half-angle hand sweeps around; watch the chosen sign flip exactly as crosses an axis. when lands in Quadrant I or II (upper half) — there . when lands in Quadrant III or IV (lower half) — there . when lands in Quadrant I or IV (right half) — there . when lands in Quadrant II or III (left half) — there . Use the quadrant of , never the quadrant of . If (Quadrant IV), then sits in Quadrant II — so is still positive . Put it to work. Since , the cosine half-angle formula turns a value you know — — into the exact value of . Play the animation: the half-angle lands in Quadrant I, fixing the sign, then the substitution resolves.

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