## Trigonometric Identities and Formulas

There are three systems of measuring Angles-

### 1. Circular System

• Unit of Measurement is radian.
• 180 degree = π radians

### 2. Sexagesimal System (English System)

• Right angle is divided into 90 equal parts called degree.
• Unit of Measurement is degree.
• Each degree is divided into 60 equal parts called minute. (1 degree = 60’)
• Each minute divided into 60 equal parts called seconds (1 minute = 60’’)

### 3. Centesimal or French System

• Right angle is divided into 100 equal parts.
• Unit of measurement is grades.
• Each grade is divided into 100 equal parts called minute, and minutes into seconds.

Sign Conventions

• cos (90 – θ) = sinθ
• tan (90 – θ) = cotθ
• cosec (90 – θ) = secθ
• sec (90 – θ) = cosecθ
• cot (90 – θ) = tanθ

### Some Basic Formulas and Identities

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

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) = (tan A + tan B) / (1 – tan A tan B)

tan (A – B) = (tan A – tan B) / (1+ tan A tan B)

cot (A + B) = (cot A cot B – 1) / (cot A + cot B)

cot (A – B) = (cot A cot B + 1) / (cot B – cot A)

sin 2θ = 2sinθcosθ

2sin A sin B = cos (A – B) – cos (A + B)

2cos A cos B = cos (A + B) + cos (A – B)

2sin A cos B = sin (A + B) + sin (A – B)

2cos A sin B = sin (A + B) – sin (A – B)

sinC + sinD = 2sin[(C + D)/2] × cos[(C – D)/2]

sinC – sinD = 2cos[(C + D)/2] × sin[(C – D)/2]

cosC + cosD = 2cos[(C + D)/2]cos[(C – D)/2]

cosC – cosD = 2sin[(C + D)/2]cos[(D – C)/2]

$\tan { (A+B+C) } =\frac { \tan { A } +\tan { B } +\tan { C } -(\tan { A } \tan { B } \tan { C } ) }{ 1-\tan { A } \tan { B } -\tan { B } \tan { C } -\tan { C } \tan { A } }$

$\sin{ 2\theta =\frac{2\tan{2\theta}}{1+\tan ^{ 2 }{\theta}}}$

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

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

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

$\cos { 2\theta } =\frac { 1-\tan ^{ 2 }{ \theta } }{ 1+\tan ^{ 2 }{ \theta } }$

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

$\sin { 3\theta } =3\sin { \theta } -4\sin ^{ 3 }{ \theta }$

$\cos {3\theta}=4\cos ^{ 3 }{ \theta} -3\cos {\theta}$

$\tan { 3\theta }=\frac { 3\tan { \theta } -\tan ^{ 3 }{ \theta } }{ 1-3\tan ^{ 2 }{ \theta } }$