Double-Angle Formulas

Pre-Calculus · Axiom Academy

Build , , and straight from the angle-sum formulas — and watch each one double. 1. The Sine Double-Angle Formula Start from the sine angle-sum formula and feed it the same angle twice. Everything collapses into a single clean product. 2. The Cosine Double-Angle Formulas The same trick on cosine gives one formula — and the Pythagorean identity rewrites it into two more. All three are the same number ; you pick whichever matches what you already know. Swap (or the reverse) — the value never changes. 3. The Tangent Double-Angle Formula Tangent's angle-sum formula, with both angles equal to , hands you the tangent of the doubled angle directly. If and is in Quadrant I, find and . First find from the Pythagorean identity (positive in Quadrant I): Now apply the double-angle formulas: One input angle drives every double-angle output together. Watch , , and move as sweeps — they are all just functions of that one angle. You've derived the double-angle formulas from the angle-sum formulas, seen the three faces of , and used them on a real value. Scroll up to revisit any step.

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