Product to Sum Calculator

This calculator evaluates the three product to sum trigonometric identities for any two angles, giving you the value of sin A sin B, cos A cos B and sin A cos B in one step, expressed as an equivalent sum or difference of cosines or sines. You enter Angle A and Angle B, choose degrees or radians, and the calculator instantly returns all three products: sin A sin B as the main result, with cos A cos B and sin A cos B alongside it, updating live as you change either angle or switch units. For example, sin A sin B equals half of cos(A minus B) minus cos(A plus B), and the other two products follow the matching identities set out below. These identities, historically called prosthaphaeresis, turn a multiplication of trig functions into an addition, a shortcut used before logarithms existed and still applied in signal processing, acoustics and Fourier analysis, where products of sines and cosines are routinely rewritten as sums. Use the calculator to check homework, verify a derivation, or evaluate a product quickly without working through the identity by hand. The worked example below, using 30 and 45 degrees, shows how to confirm the maths yourself, and angles are converted to radians internally before the trig functions are applied, with results rounded for display.

Calculate.co.nz is proud to be partnered with Health Based Building, a leader in sustainable and health-conscious building innovation. With over a century of experience, they develop high-performance systems like Foreverbreathe Specification, Magnum Board, and Foreverbreathe Paints to support energy-efficient, non-toxic living environments. Their commitment to healthier homes aligns with our belief that informed choices lead to better outcomes for Kiwi households.