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## Trigonometry

### Course: Trigonometry>Unit 1

Lesson 7: The reciprocal trigonometric ratios

# 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.
sine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis
Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90degrees.
Now here's the cool part. See how the sine of one acute angle
describes the start color #11accd, start text, e, x, a, c, t, space, s, a, m, e, space, r, a, t, i, o, end text, end color #11accd as the cosine of the other acute angle?
Incredible! Both functions, sine, left parenthesis, theta, right parenthesis and cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis, give the exact same side ratio in a right triangle.
And we're done! We've shown that sine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis.
In other words, the sine of an angle equals the cosine of its complement.
Well, technically we've only shown this for angles between 0degrees and 90degrees. 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, left parenthesis, theta, right parenthesis, equals, g, left parenthesis, 90, degrees, minus, theta, right parenthesis
and
g, left parenthesis, theta, right parenthesis, equals, f, left parenthesis, 90, degrees, minus, theta, right parenthesis.
Here is a full list of the basic trigonometric cofunctions:
Cofunctions
Sine and cosinesine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis
cosine, left parenthesis, theta, right parenthesis, equals, sine, left parenthesis, 90, degrees, minus, theta, right parenthesis
Tangent and cotangenttangent, left parenthesis, theta, right parenthesis, equals, cotangent, left parenthesis, 90, degrees, minus, theta, right parenthesis
cotangent, left parenthesis, theta, right parenthesis, equals, tangent, left parenthesis, 90, degrees, minus, theta, right parenthesis
Secant and cosecant\sec, left parenthesis, theta, right parenthesis, equals, \csc, left parenthesis, 90, degrees, minus, theta, right parenthesis
\csc, left parenthesis, theta, right parenthesis, equals, \sec, left parenthesis, 90, degrees, minus, theta, right parenthesis
Neat! Whoever named the trig functions must have deeply understood the relationships between them.