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(θ)=cos(90θ)\sin(\theta) = \cos(90^\circ-\theta)
It can be fun to verify that this property works on your calculator. Try plugging in sin(10)\sin(10^\circ) and cos(80)\cos(80^\circ), and check that both give you the same answer: 0.17364817766.
Note: Make sure your calculator is in degrees mode, not radians.
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 θ\theta, then the other must be 90^\circ minus θ\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 exact same ratio\blueD{\text{exact same ratio}} as the cosine of the other acute angle?
Incredible! Both functions, sin(θ)\sin(\theta) and cos(90θ)\cos(90^\circ-\theta), give the exact same side ratio in a right triangle.
And we're done! We've shown that sin(θ)=cos(90θ)\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.


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 ff and gg are cofunctions, then
f(θ)=g(90θ)f(\theta) = g(90^\circ-\theta)
g(θ)=f(90θ)g(\theta) = f(90^\circ-\theta).
Here is a full list of the basic trigonometric cofunctions:
Sine and cosinesin(θ)=cos(90θ)\sin(\theta) = \cos(90^\circ-\theta)
cos(θ)=sin(90θ)\cos(\theta) = \sin(90^\circ-\theta)
Tangent and cotangenttan(θ)=cot(90θ)\tan(\theta) = \cot(90^\circ-\theta)
cot(θ)=tan(90θ)\cot(\theta) = \tan(90^\circ-\theta)
Secant and cosecantsec(θ)=csc(90θ)\sec(\theta) = \csc(90^\circ-\theta)
csc(θ)=sec(90θ)\csc(\theta) = \sec(90^\circ-\theta)
Neat! Whoever named the trig functions must have deeply understood the relationships between them.