Trigonometry Formulas

Trigonometric Ratios

Given a right triangle with angle  `\theta`

Sine Ratio:   `\sin  \theta = \frac{\text{Opposite side} }{\text{Hypotenuse}} `

Cosine Ratio:  `\cos \theta = \frac{\text{Adjacent side} }{\text{Hypotenuse}} `

Tangent Ratio: `\tan  \theta = \frac{\text{Opposite side} }{\text{Adjacent side}} `

Cosecant Ratio: `\csc  \theta = \frac{\text{Hypotenuse} }{\text{Opposite side}} `

Secant Ratio : `\sec  \theta = \frac{\text{Hypotenuse} }{\text{Adjacent side}} `

Cotangent Ratio : `\cot  \theta = \frac{\text{Adjacent side} }{\text{Opposite side}} `

Trigonometric Identities

Reciprocal Identities

`\sin \theta = \frac{1}{\csc \theta}`

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

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

`\csc \theta = \frac{1}{\sin \theta}`

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

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

Tangent and Cotangent Identities

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

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

Pythagorean Identities

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

` 1+ \tan^{2} \theta = \csc^{2}\theta ` 

`1+\cot^{2}\theta =\sec^{2}\theta `

Double Angle Formulas

`\sin 2\theta = 2 \sin theta \cos theta`

`\cos 2 \theta = \cos^2 \theta- \sin^2 \theta`

`\cos 2 \theta = 1-2 \sin^2 \theta`

`\cos 2 \theta =2\cos^2 \theta - 1`

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

Even and Odd Angle Formulas

`\sin(-θ) = -\sinθ

$$\cos(-θ) = \cosθ$$ 

$$\tan(-θ) = -\tanθ$$

 $$\cot(-θ) = -\cotθ$$

 $$\sec(-θ) = \secθ$$

 $$\csc(-θ) = -\cscθ$$

Triple Angle Formulas

`\sin(3\theta)=3 \sin{\theta}-4\sin^3 (\theta)`

`\cos(3\theta)=4cos^3(\theta)-3 cos (\theta )`

`\tan(3\theta)= \frac{3\tan \theta -\tan^3\theta}{1-3\tan^2\theta}`

Addition Trigonometric 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}`

Factorisation Formulas

`\sin A +\sin B =2 \sin(\frac{A+B}{2}\cos(\frac{A-B}{2})`

`\sin A - \sin B = 2\cos (\frac{A+B}{2})\sin(\frac{A-B}{2})`

`\cos A + \cos B = 2 \cos(\frac{A+B}{2})\cos (\frac{A-B}{2}) `

`\cos A - \cos B = -2 \sin (\frac{A+B}{2})\sin (\frac{A-B}{2})`
 

Trigonometry Table