Double-Angle Formulas

Trigonometry · Axiom Academy

Master the formulas for sin(2x), cos(2x), and tan(2x), and learn when to use each form of cos(2x) effectively. We start with the sum formula for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). What happens when we let A = B = x? Starting from the sum formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can derive cos(2x). But here's where it gets interesting: we can rewrite this formula in three different ways using the Pythagorean identity sin²(x) + cos²(x) = 1. Each form of cos(2x) is strategically useful in different situations. Choosing the right form can dramatically simplify your work! For tangent, we use tan(A + B) = [tan(A) + tan(B)] / [1 - tan(A)tan(B)]. Setting A = B = x gives us the double-angle formula for tangent. Note: This formula is undefined when tan²(x) = 1, which occurs at x = π/4 + nπ/2.

This is the written version of the interactive lesson above. See the full Trigonometry course.