Cofunction Identities

Trigonometry · Axiom Academy

Why every function equals the co-function of the complementary angle — — seen three ways. 1. Complementary Angles Share Their Sides Every right triangle has one angle, so the other two must add to — they are complementary . Now watch a single leg: it is the side opposite one acute angle and, at the very same time, the side adjacent to the other. the sine of an angle is the cosine of its complement and the cosine of an angle is the sine of its complement 2. Reflecting the Unit Circle Across y = x On the unit circle the point at angle has coordinates . Reflecting it across the line y = x lands it at angle and simply swaps the two coordinates into . Reading the swapped coordinates back off the circle produces every cofunction identity at once. The same six pairings hold in radians, with written as the quarter-turn above. 3. Reading a Value Off the Complement Across to the sine and cosine graphs are mirror images about . So the height of sine at equals the height of cosine at : you can trade any value for its cofunction at the complementary angle, with no new calculation. You've seen the cofunction identities three ways — and why every function equals the co-function of the complementary angle. Scroll up to revisit any step.

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