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:

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

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

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

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