Complete List of Trigonometry Formulas for CGL CHSL Exams

Trigonometric Formulas:

Applying Pythagoras theorem for the given right-angled theorem, we have:

$(Perpendicular{)}^2+(Base{)}^2=(Hypotenuese{)}^2$

$\Rightarrow (P{)}^2+(B{)}^2=(H{)}^2$

The Trigonometric properties are given below:

S.no	Property	Mathematical value
1	$\sin A$	$\frac{P}{H}$
2	$\cos A$	$\frac{B}{H}$
3	$\tan A$	$\frac{P}{B}$
4	$\cot A$	$\frac{B}{P}$
5	$cosecA$	$\frac{H}{P}$
6	$\sec A$	$\frac{H}{B}$

Relation Between Trigonometric Identities:

S.no	Identity	Relation
1	$\tan A$	$\frac{\sin A}{\cos A}$
2	$\cot A$	$\frac{\cos A}{\sin A}$
3	$cosecA$	$\frac{1}{\sin A}$
4	$\sec A$	$\frac{1}{\cos A}$

Trigonometric Identities:

1. ${\sin }^{2}A+{\cos }^{2}A=1$

2. ${\tan }^{2}A+\; 1\; =sec{}^{}A$

3. ${\cot }^{2}A+\; 1\; =cose{c}^{2}A$

Trigonometry Negative Formulas

$\sin (-\theta )=-\sin \theta$

$\cos (-\theta )=\cos \theta$

$\tan (-\theta )=-\tan \theta$

$cosec(-\theta )=-cosec\theta$

$\sec (-\theta )=\sec \theta$

$\cot (-\theta )=-\cot \theta$

Product to Sum Formulas

$\sin xsiny=\frac{1}{2}\left[\cos \left(x–y\right)-\cos \left(x+y\right)\right]$

$\cos x\cos y=\frac{1}{2}\left[\cos \left(x–y\right)+\cos \left(x+y\right)\right]$

$\sin x\cos y=\frac{1}{2}\left[\sin \left(x+y\right)+\sin \left(x-y\right)\right]$

$\cos x\sin y=\frac{1}{2}\left[\sin \left(x+y\right)–\sin \left(x-y\right)\right]$

Sum to Product Formulas

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

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

$\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$\cos x-\cos y=–2\sin \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$

Basic Formulas

$\sin (A+B)=\sin A\cos B+\cos A\sin B$ $\sin (A-B)=\sin A\cos B–\cos A\sin B$ $\cos (A+B)=\cos A\cos B–\sin A\sin B$ $\cos (A–B)=\cos A\cos B+\sin A\sin B$ $\tan (A+B)=\frac{\tan A+\tan B}{1–\tan A\tan B}$ $\tan (A–B)=\frac{\tan A–\tan B}{1+\tan A\tan B}$ $\cos (A+B)\cos (A–B)={\cos }^{2}A–{\sin }^{2}B={\cos }^{2}B–{\sin }^{2}A$ $\sin (A+B)\sin (A–B)={\sin }^{2}A–{\sin }^{2}B={\cos }^{2}B–{\cos }^{2}A$ $\sin 2A=2\sin A\cos A=\frac{2\tan A}{1+{\tan }^{2}A}$ $\cos 2A={\cos }^A–{\sin }^{2}A=1–2si{n}^{2}A=2co{s}^{2}A–1=\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}$ $\tan 2A=\frac{2\tan A}{1–{\tan }^{2}A}$ $\cos 3A=4{\cos }^{3}A–3\cos A=4\cos \left({60}^{\circ }-A\right).\cos A.\cos \left({60}^{\circ }+A\right)$ $\tan 3A=\frac{3\tan A–{\tan }^{3}A}{1-3{\tan }^{2}A}=\tan \left({60}^{\circ }-A\right).\tan A.\tan \left({60}^{\circ }+A\right)$ $\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$

Definition

$\theta ={\sin }^{-1}\left(x\right)isequivalenttox=\sin \theta$

$\theta ={\cos }^{-1}\left(x\right)isequivalenttox=\cos \theta$

$\theta ={\tan }^{-1}\left(x\right)isequivalenttox=\tan \theta$

Inverse Properties

$\sin \left({\sin }^{-1}\left(x\right)\right)=x$

$\cos \left({\cos }^{-1}\left(x\right)\right)=x$

$\tan \left({\tan }^{-1}\left(x\right)\right)=x$

${\sin }^{-1}\left(\sin \left(\theta \right)\right)=\theta$

${\cos }^{-1}\left(\cos \left(\theta \right)\right)=\theta$

${\tan }^{-1}\left(\tan \left(\theta \right)\right)=\theta$

Double Angle and Half Angle Formulas

$\sin \left(2x\right)=2\sin x\cos x$

$\cos \left(2x\right)={\cos }^{2}x–{\sin }^{2}x$

$\tan \left(2x\right)=\frac{2\tan x}{1–{\tan }^{2}x}$

$\sin \frac{x}{2}=\pm \sqrt{\frac{1–\cos x}{2}}$

$\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}$

$\tan \frac{x}{2}=\frac{1-\cos x}{\sin x}=\frac{\sin x}{1+\cos x}$