A **cofunction **is a complementary function that typically describes two functions with complementary angles. Cofunctions can be applied to all of the trigonometric functions. For example:

Notice the “co” prefix: that indicates the function is a cofunction. This terminology dates back to 1620, when Edmund Gunter used the word *co.sinus* in his *Canon triangulorum*. The word was modified by John Newton (1658) into *cosinus *although the word *cosine *also appeared in a 1635 text by John Wells [1].

Here is a more formal definition:

If a function f is a cofunction of *g*, then the output of function *g* when evaluated at a particular angle will be equal to the output of f when evaluated at the angle complementary to that (complementary angles are two angles whose sum is 90°).

So if *f* is a cofunction of *g*, f(A) = g(B) whenever A and B are complementary angles.

## Examples of Cofunction Relationships

You can see the cofunction identities in action if you plug a few values for sine and cosine into your calculator.

- The sine of ten° is 0.17364817766683; and this is exactly the same as the cosine of 80°. Note that 10° + 80° = 90°; they are complementary angles.
- The sine of 20° is 0.34202014332567; which is the same as the cosine of 70°.
- The sine of 30° is 0.5, as is cosine of 60°.

You can see this graphically if you graph the two cofunctions, sine and cosine, on the same graph. They look like the same function, just shifted along the x axis.

## Unpacking Complementary Angles

Complementary angles are angle pairs whose sum adds up to a right angle: 90°, or π/2 radians. That makes the complementary angle to a 20 degree angle:

90 – 20 = 70 (because 70 + 20 = 90).

Similarly, the complementary angle of a π/6 angle is π/3 = (2π)/6, because (2π)/6 + π/6 = (3π)/6 = π/2.

## The Cofunction Identities

The cofunction identities simply tell us the relationships between sine, tangent, and secant and their cofunctions:

## References

[1] Earliest Known Uses of Some of the Words of Mathematics (C).

Sum and Difference Formulas. Retrieved from https://www.alamo.edu/contentassets/35e1aad11a064ee2ae161ba2ae3b2559/analytic/math2412-sum-difference-indentities.pdf on January 27, 2018.

Summary of Trigonometric Identities. Retrieved from https://www2.clarku.edu/faculty/djoyce/trig/identities.html on January 27, 2018.