Types of Trigonometric Functions

$\begin{aligned} & \sin \theta=\frac{\text { Perpendicular }}{\text { Hypotenuse }}=\frac{\mathrm{BC}}{\mathrm{AC}} \\ & \cos \theta=\frac{\text { Base }}{\text { Hypotenuse }}=\frac{\mathrm{AB}}{\mathrm{AC}} \\ & \tan \theta=\frac{\text { Perpendicular }}{\text { Base }}=\frac{\mathrm{BC}}{\mathrm{AB}} \\ & \operatorname{cosec}=\frac{1}{\sin \theta} \\ & \sec =\frac{1}{\cos \theta} \\ & \cot =\frac{\cos \theta}{\sin \theta}=\frac{1}{\tan \theta}\end{aligned}$

The above circle is the standard unit circle (centre at the origin and radius is equal to 1 unit)

$\begin{aligned} & \sin \theta=\frac{x}{r}=\frac{x}{1}=1 \\ & \cos \theta=\frac{y}{r}=\frac{y}{1}=1\end{aligned}$

For a standard unit circle, the value of x and y give us the values of $\cos \theta$ and $\sin \theta$ respectively.

Graphs of trigonometric functions:

Trigonometric Identities-1 (Reciprocal ,Pythagorean ,Co-function and Even-Odd Identities)

Trigonometric Identities:-

• Reciprocal Identities

$\begin{aligned} & \csc \theta=\frac{1}{\sin \theta} \\ & \sec \theta=\frac{1}{\cos \theta} \\ & \cot \theta=\frac{1}{\tan \theta}\end{aligned}$

• Pythagorean Identities

$\begin{aligned} & \sin ^2 \theta+\cos ^2 \theta=1 \\ & \sec ^2 \theta=1+\tan ^2 \theta \\ & \csc ^2 \theta=1+\cot ^2 \theta\end{aligned}$

• Co-function Identities

$\begin{aligned} & \cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta \\ & \sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta \\ & \cot \left(\frac{\pi}{2}-\theta\right)=\tan \theta \\ & \tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta \\ & \csc \left(\frac{\pi}{2}-\theta\right)=\sec \theta \\ & \sec \left(\frac{\pi}{2}-\theta\right)=\csc \theta\end{aligned}$

• Even-Odd Identities

$\begin{aligned} & \sin (-\theta)=-\sin \theta \\ & \cos (-\theta)=\cos \theta \\ & \tan (-\theta)=-\tan \theta \\ & \csc (-\theta)=-\csc \theta \\ & \sec (-\theta)=\sec \theta \\ & \cot (-\theta)=-\cot \theta\end{aligned}$

Trigonometric Identities-2 ( Double angled,half angled and Triple angled formulas)

Trigonometric Identities:-

• Double angled formulas

$\begin{aligned} \sin 2 \theta & =2 \sin \theta \cos \theta \\ \cos 2 \theta & =\cos ^2 \theta-\sin ^2 \theta \\ & =2 \cos ^2 \theta-1 \\ & =1-2 \sin ^2 \theta \\ \tan 2 \theta & =\frac{2 \tan \theta}{1-\tan ^2 \theta}\end{aligned}$

• Half angled formulas

$\begin{aligned} & \sin \left(\frac{\theta}{2}\right)= \pm \sqrt{\frac{1-\cos \theta}{2}} \\ & \cos \left(\frac{\theta}{2}\right)= \pm \sqrt{\frac{1+\cos \theta}{2}} \\ & \tan \left(\frac{\theta}{2}\right)=\frac{1-\cos \theta}{\sin \theta}\end{aligned}$

• Triple angled identities

$\begin{aligned} & \sin 3 x=3 \sin x-4 \sin ^3 x \\ & \cos 3 x=4 \cos ^3 x-3 \cos x \\ & \tan 3 \mathrm{x}=\frac{3 \tan x-\tan ^3 x}{1-3 \tan ^2 x}\end{aligned}$

Trigonometric Identities-3 (Sum and difference identities, Sum-to-Product and Product-to-Sum Formulas)

Trigonometric Identities:-

• Sum and difference identities

$\begin{aligned} & \sin (x+y)=\sin x \cos y+\cos x \sin y \\ & \sin (x-y)=\sin x \cos y-\cos x \sin y \\ & \cos (x+y)=\cos x \cos y-\sin x \sin y \\ & \cos (x-y)=\cos x \cos y+\sin x \sin y \\ & \tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y} \\ & \tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}\end{aligned}$

• Sum-to-Product Formulas

$\begin{aligned} & \sin x+\sin y=2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) \\ & \sin x-\sin y=2 \cos \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right) \\ & \cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) \\ & \cos x-\cos y=-2 \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)\end{aligned}$

• Product-to-Sum Formulas

$\begin{aligned} \sin x \sin y & =\frac{1}{2}[\cos (x-y)-\cos (x+y)] \\ \cos x \cos y & =\frac{1}{2}[\cos (x-y)+\cos (x+y)] \\ \sin x \cos y & =\frac{1}{2}[\sin (x+y)+\sin (x-y)] \\ \cos x \sin y & =\frac{1}{2}[\sin (x+y)-\sin (x-y)]\end{aligned}$

Trigonometric Ratios of Allied Angles

Trigonometric ratios of allied angles: