# Trigonometry Functions, Formulas and Identities

Posted in Terms Defined

Please refer to the following figure of right-angled triangle:

Trigonometry Functions:

$$Sine \theta = \frac{opposite}{hypotenuse} = \frac{AC}{AB}$$

$$Secant \theta = \frac{hypotenuse}{adjacent} = \frac{AB}{BC}$$

$$Cosine \theta = \frac{adjacent}{hypotenuse} = \frac{BC}{AB}$$

$$Tangent \theta = \frac{opposite}{adjacent} = \frac{AC}{BC}$$

$$Co-Secant \theta = \frac{hypotenuse}{opposite} = \frac{AB}{AC}$$

$$Co-Tangent \theta = \frac{adjacent}{opposite} = \frac{BC}{AC}$$

Reciprocal Identities:

$$Cosec \theta = \frac{1}{Sine \theta}$$

$$Secant \theta = \frac{1}{Cosine \theta}$$

$$Co-Tangent \theta = \frac{1}{Tangent \theta}$$

$$Cosine \theta = \frac{1}{Secant \theta}$$

$$Tangent \theta = \frac{1}{Co-Tangent \theta}$$

Short Forms:

$$cos \theta ==> Cosine \theta$$

$$sin \theta ==> Sine \theta$$

$$tan \theta ==> Tangent \theta$$

$$cot \theta ==> Co-Tangent \theta$$

$$sec \theta ==> Secant \theta$$

$$cosec \theta ==> Co-Secant \theta$$

Trigonometry Formulas involving Periodicity:

$$sin (\theta + 2 \pi) = sin \theta$$

$$cos (\theta + 2\pi) = cos \theta$$

$$tan (\theta + \pi) = tan \theta$$

$$cot (\theta + \pi) = cot \theta$$

Trigonometry Formulas involving Co-function Identities:

$$sin (90 - \theta ) = cos \theta$$

$$cos (90 - \theta ) = sin \theta$$

$$tan (90 - \theta = cot \theta$$

$$cot (90 - \theta = tan \theta$$

Trigonometry Formulas involving Sum / Difference Identities:

$$sin (A + B) = sin A cos B + cos A sin B$$

$$cos (A + B) = cos A cos B - sin A sin B$$

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

$$sin (A - B) = sin A cos B - cos A sin B$$

$$cos (A - B) = cos A cos B + sin A sin B$$

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

Trigonometry Formulas involving Double Angle Identities:

$$sin (2A) = 2 sin (A) cos (A)$$

$$cos (2A) = cos^2 (A) - sin^2 (A)$$

$$sin^2 (A) + cos^2 (A) = 1$$

$$cos (2A) = 2 cos^2 (A) - 1$$

$$cos (2A) = 1 - 2 sin^2 (A)$$

$$tan (2A) = \frac{2 tan (A)}{ 1 - tan^2(A)}$$

Trigonometry Formulas involving Half Angle Identities:

$$sin (\frac{A}{2}) = \sqrt{\frac{1 - cos A}{2}}$$

$$cos (\frac{A}{2}) = \sqrt{\frac{1 + cos A}{2}}$$

$$tan (\frac{A}{2}) = \sqrt{\frac{1 - cos A}{1 + cos A}}$$

$$tan (\frac{A}{2}) = \frac{1 - cos A}{sin A}$$

Trigonometry Formulas involving Product Identities:

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

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

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

Trigonometry Formulas involving Sum to Product Identities:

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

$$sin A - sin B = 2 cos (\frac{A+B}{2} sin (\frac{x-y}{2})$$

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

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