Products of Sines and Cosines

Calculus 2 · Axiom Academy

EXAMPLE Products of Sines and Cosines Integrate a product of trigonometric functions using a product-to-sum identity. Evaluate the integral . The integrand is a product of a sine and a cosine with different arguments, so we cannot integrate it directly — we first rewrite it as a sum. Nice work — you integrated a product of trigonometric functions by turning it into a sum. Here's what carried the solution: Product-to-sum identities: they convert products of sines and cosines into sums, which integrate term by term. Integration rule: after splitting into a sum, . Mind the coefficients: the from the identity and the from each integral combine — distribute carefully. The same product-to-sum trick handles and too, and it's the workhorse behind Fourier analysis and signal processing.

This is the written version of the interactive lesson above. See the full Calculus 2 course.