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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.
Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90.
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(90θ), give the exact same side ratio in a right triangle.
And we're done! We've shown that sin(θ)=cos(90θ).
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 cosinesin(θ)=cos(90θ)
Tangent and cotangenttan(θ)=cot(90θ)
Secant and cosecantsec(θ)=csc(90θ)
Neat! Whoever named the trig functions must have deeply understood the relationships between them.

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