Sometimes, expressing a function in terms of its co-functions helps solve the problem easily. So, what are **co-functions**, and how do they change the ratios? These can be confusing. Let’s see how this works:

For example, expressing **$\sin x$ as $\cos (\large\frac{\pi}{2}$$-x)$** might help solve the problem more easily than if you left it as $\sin x$.

This brings us to the concept of **co-functions**. Wikipedia helps us w/ this friendly definition: Whenever *A* and *B* are **complementary angles, ** a function f is a co-function of a function g if *f(A) = g(B). *

So what are **complementary angles**? Complementary angles are angles whose sum = $90 ^{\circ}$ (or $270 ^{\circ}$) degrees. Therefore:

- $\sin x$ = $\cos (\large\frac{\pi}{2}$$-x)$
- $\cos x$ = $\tan (\large\frac{\pi}{2}$$-x)$
- $\tan x$ = $\cot (\large\frac{\pi}{2}$$-x)$
- $\cot x$ = $\tan (\large\frac{\pi}{2}$$-x)$
- $\text{cosec } x$ = $\sec (\large\frac{\pi}{2}$$-x)$
- $\sec x$ = $ \text{cosec} (\large\frac{\pi}{2}$$-x)$

We can see that this change in the ratios takes place only along the y-axis, i.e; along 90 and 270 degrees and the ratios remain the same along the x-axis. That is along 0 and 180 degrees.

This, when combined with 'All Silver tea cups", makes it easy to remember the ratios as well as the sign in the respective quadrants.

Here are a couple of problems that use cofuction: http://clay6.com/qa/1048 and http://clay6.com/qa/3016