# Cofunction Formulas

We often come across with functions in mathematics. A function f is co-function of a function g if f(A) = g(B) whenever A and B are complementary angles.

A mathematical function is said to be a special kind of relation between inputs and outputs, where every input value is connected with exactly one output value by the means of a particular property. The trigonometric functions are one of the most important ones.

## Co-function Identities

$\large \sin \theta =\cos\:(90-\theta)$ $\large \cos \theta =\sin\:(90-\theta)$ $\large \tan\theta =\cot\:(90-\theta)$ $\large \csc \theta =\sec\:(90-\theta)$ $\large \sec\theta =\csc\:(90-\theta)$ $\large \cot\theta =\tan\:(90-\theta)$

## Formula of Co-function

$\large \sin \left [\frac{\pi}{2}-\theta\right]=\cos\theta$ $\large \cos \left [\frac{\pi}{2}-\theta\right]=\sin \theta$ $\large \tan\left [\frac{\pi}{2}-\theta\right]=\cot \theta$ $\large \cot\left [\frac{\pi}{2}-\theta\right]=\tan\theta$ $\large \csc\left [\frac{\pi}{2}-\theta\right]=\sec\theta$ $\large \sec\left [\frac{\pi}{2}-\theta\right]=\csc\theta$

Trigonometric Co-Functions

 $\large \sin\theta$ $\large \cos\theta$ $\large \tan\theta$ $\large \csc\theta$ $\large \sec\theta$ $\large \cot\theta$ $\large \sin\theta$ $\large \sin\theta$ $\large \pm \sqrt{1-\cos^{2}\theta}$ $\large \pm \frac{\tan\theta}{\sqrt{1+\tan ^{2}\theta}}$ $\large \frac{1}{\csc\theta}$ $\large \pm \frac{\sqrt{\sec^{2}-1}}{\sec\theta}$ $\large \pm \frac{1}{\sqrt{1+\cot^{2}\theta}}$ $\large \cos\theta$ $\large \pm \sqrt{1-\sin ^{2}\theta}$ $\large \cos\theta$ $\large \pm \frac{1}{\sqrt{1+\tan^{2}\theta}}$ $\large \pm \frac{\sqrt{\csc^{2}\theta-1}}{\csc\theta}$ $\large \frac{1}{\sec\theta}$ $\large \pm \frac{\cot\theta}{\sqrt{1+\cot^{2}\theta}}$ $\large \tan\theta$ $\large \pm \frac{\sin\theta}{\sqrt{1-\sin ^{2}}\theta}$ $\large \pm \frac{\sqrt{1-\cos^{2}\theta}}{\cos\theta}$ $\large \tan\theta$ $\large \pm \frac{1}{\sqrt{\csc^{2}\theta-1}}$ $\large \pm \sqrt{\sec^{2}\theta -1}$ $\large \frac{1}{\cot\theta}$ $\large \csc\theta$ $\large \frac{1}{\sin\theta}$ $\large \pm \frac{1}{\sqrt{1-\cos ^{2}\theta}}$ $\large \pm \frac{\sqrt{1+\tan ^{2}\theta}}{\tan\theta}$ $\large \csc\theta$ $\large \pm \frac{\sec\theta}{\sqrt{\sec^{2}\theta-1}}$ $\large \pm{\sqrt{1+\cot^{2}\theta}}$ $\large \sec\theta$ $\large \pm \frac{1}{\sqrt{1-\sin^{2}\theta}}$ $\large \frac{1}{\cos\theta}$ $\large \pm{\sqrt{1+\tan^{2}\theta}}$ $\large \pm \frac{\csc\theta}{\sqrt{\csc^{2}\theta-1}}$ $\large \sec\theta$ $\large \pm \frac{\sqrt{1+\cot^{2}\theta}}{\cot\theta}$ $\large \cot\theta$ $\large \pm \frac{\sqrt{1-\sin^{2}\theta}}{\sin\theta}$ $\large \pm \frac{\cos\theta}{\sqrt{1-\cos^{2}\theta}}$ $\large \frac{1}{\tan\theta}$ $\large \pm{\sqrt{\csc^{2}\theta-1}}$ $\large \pm \frac{1}{\sqrt{\sec^{2}\theta -1}}$ $\large \cot\theta$

#### Practise This Question

The figure consists of 2 cylinders, the inner cylinder is a solid cylinder whose radius is r and the outer cylinder is a hollow cylinder whose radius is R and height is h, the volume of fluid it can hold is: