Cofunction Identity
Sine and cosine of complementary angles are equal.
Formula
Variables
Example
sin30°=cos60°
How it works
This identity says that if two acute angles add up to 90°, the sine of one angle matches the cosine of the other. It works because in a right triangle, the side opposite one acute angle is the same side adjacent to the complementary angle, so the ratio stays the same. In unit-circle terms, swapping an angle with its complement changes the horizontal and vertical coordinates, which turns cosine into sine.
When to use it
Use this when you need to rewrite a sine as a cosine, or a cosine as a sine, especially with complementary angles in right-triangle trigonometry. It is also useful for simplifying trig expressions and checking exact values without a calculator.
Worked example, step by step
Find sin 20° using a complementary angle. sin 20° = cos(90° - 20°) sin 20° = cos 70° If cos 70° ≈ 0.3420, then sin 20° ≈ 0.3420 Final answer: sin 20° = cos 70° ≈ 0.3420
Common mistakes to avoid
- Using 90° + θ instead of 90° - θ
- Forgetting that the identity works for complementary angles
- Mixing up which function gets rewritten: sine becomes cosine, and cosine becomes sine
Did You Know?
Sine and cosine are cofunctions because the sine of an angle equals the cosine of its complement — hence the "co".
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