The angles in a triangle add up to 180∘. Since one angle in a right triangle measures 90∘, the remaining two angles must add up to 180∘ minus 90∘, or 90∘.
So, if we name one of the angles θ, then the other must be 90∘ minus θ. This ensures that the two angles sum to 90∘.
When two angles sum to 90∘, we call them complementary angles.
Now here's the cool part. See how the sine of one acute angle
describes the exact same ratio as the cosine of the other acute angle?
Incredible! Both functions, sin(θ) and cos(90∘−θ), give the exact same side ratio in a right triangle.
And we're done! We've shown that sin(θ)=cos(90∘−θ).
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∘ and 90∘. 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 f and g are cofunctions, then
Here is a full list of the basic trigonometric cofunctions:
Sine and cosine
Tangent and cotangent
Secant and cosecant
Neat! Whoever named the trig functions must have deeply understood the relationships between them.