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What are co-functions and how can we simplify trigonometric problems by using co-functions?

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

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