Question

Different Formulae in Trigonometry

Answer 1

Here, some basic trigonometric formulae are given:

$\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}$

$\cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}$

$\tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}$

$\cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}$

$\sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}$

$\text{cosec}\theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}$

Reciprocals of trigonemetric ratios:

$\sin \theta \times \text{cosec}\theta =1$

$\Rightarrow \sin \theta =\frac{1}{\text{cosec}\theta }$

$\text{cosec}\theta =\frac{1}{\sin \theta }$

$\cos \theta \times \sec \theta =1$

$\Rightarrow \cos \theta =\frac{1}{\sec \theta }$

$\sec \theta =\frac{1}{\cos \theta }$

$\tan \theta \times \cot \theta =1$

$\Rightarrow \tan \theta =\frac{1}{\cot \theta }$

$\cot \theta =\frac{1}{\tan \theta }$

Here,

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

$\cot \theta =\frac{\cos \theta }{\sin \theta }$

Trigonometric identities:

${\sin }^2\theta +{\cos }^2\theta =1$

$\Rightarrow {\sin }^2\theta =1-{\cos }^2\theta$

${\cos }^2\theta =1-{\sin }^2\theta$

${\sec }^2\theta -{\tan }^2\theta =1,\theta \ne (2n+1)\frac{\pi }{2},n\in Z$

$\Rightarrow {\sec }^2\theta =1+{\tan }^2\theta$

${\tan }^2\theta ={\sec }^2\theta -1$

${\text{cosec}}^2\theta -{\cot }^2\theta =1,\theta \ne n\pi ,n\in Z$

$\Rightarrow {\text{cosec}}^2\theta =1+{\cot }^2\theta$

$\Rightarrow {\cot }^2\theta ={\text{cosec}}^2\theta -1$