# Sine & cosine of complementary angles

Learn about the relationship between the sine & cosine of complementary angles, which are angles who together sum up to 90°.
We want to prove that the sine of an angle equals the cosine of its complement.
$\sin(\theta) = \cos(90^\circ-\theta)$
It can be fun to verify that this property works on your calculator. Try plugging in $\sin(10^\circ)$ and $\cos(80^\circ)$, and check that both give you the same answer: 0.17364817766.
Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90$^\circ$.
The angles in a triangle add up to 180$^\circ$. Since one angle in a right triangle measures 90$^\circ$, the remaining two angles must add up to 180$^\circ$ minus 90$^\circ$, or 90$^\circ$.
So, if we name one of the angles $\theta$, then the other must be 90$^\circ$ minus $\theta$. This ensures that the two angles sum to 90$^\circ$.
When two angles sum to 90$^\circ$, we call them complementary angles.
Now here's the cool part. See how the sine of one acute angle
describes the $\blueD{\text{exact same ratio}}$ as the cosine of the other acute angle?
Incredible! Both functions, $\sin(\theta)$ and $\cos(90^\circ-\theta)$, give the exact same side ratio in a right triangle.
And we're done! We've shown that $\sin(\theta) = \cos(90^\circ-\theta)$.
In other words, the sine of an angle equals the cosine of its complement.
Well, technically we've only shown this for angles between 0$^\circ$ and 90$^\circ$. To make our proof work for all angles, we'd need to move beyond right triangle trigonometry into the world of unit circle trigonometry, but that's a task for another time.

## Cofunctions

You may have noticed that the words sine and cosine sound similar. That's because they're cofunctions! The way cofunctions work is exactly what you saw above. In general, if $f$ and $g$ are cofunctions, then
$f(\theta) = g(90^\circ-\theta)$
and
$g(\theta) = f(90^\circ-\theta)$.
Here is a full list of the basic trigonometric cofunctions:
Cofunctions
Sine and cosine$\sin(\theta) = \cos(90^\circ-\theta)$
$\cos(\theta) = \sin(90^\circ-\theta)$
Tangent and cotangent$\tan(\theta) = \cot(90^\circ-\theta)$
$\cot(\theta) = \tan(90^\circ-\theta)$
Secant and cosecant$\sec(\theta) = \csc(90^\circ-\theta)$
$\csc(\theta) = \sec(90^\circ-\theta)$
Neat! Whoever named the trig functions must have deeply understood the relationships between them.