How do complementary angles connect sine and cosine?

In trigonometry, complementary angles are defined as two angles that sum to 90 degrees. This relationship creates a fundamental connection between the sine and cosine functions.12

In a right-angled triangle, the two acute angles are complementary because the sum of all angles in a triangle is 180 degrees, and the right angle accounts for 90 degrees. If one acute angle is denoted as $\theta$, the other must be ${90}^{\circ }-\theta$.

The connection arises because the side that is "opposite" to $\theta$ is the same side that is "adjacent" to the complementary angle (${90}^{\circ }-\theta$). Based on the ratio definitions of trigonometric functions:

• $\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos ({90}^{\circ }-\theta )=\frac{\text{adjacent to }({90}^{\circ }-\theta )}{\text{hypotenuse}}=\frac{\text{opposite to }\theta }{\text{hypotenuse}}$

Consequently, the sine of an angle is equal to the cosine of its complement, and vice versa:
* $\sin (\theta )=\cos ({90}^{\circ }-\theta )$
* $\cos (\theta )=\sin ({90}^{\circ }-\theta )$34