Trigonometry identities

Pythagorean identities

<math>\sin^2 A+\cos^2 A = 1 </math>

<math>1+\tan^2 A=\sec^2 A </math>

<math>1+\cot^2 A=\csc^2 A </math>

Law of sines

The law of sines for an arbitrary triangle states:

<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c},</math>

or equivalently:

<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R</math>

Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem to arbitrary triangles:

<math>c^2=a^2+b^2-2ab\cos C ,</math>

or equivalently:

<math>\cos C=\frac{a^2+b^2-c^2}{2ab}</math>

Sum and difference identities

<math>\sin (A + B) = \sin A \cos B + \cos A \sin B \,</math>

<math>\cos (A + B) = \cos A \cos B - \sin A \sin B \,</math>

<math>\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \,</math>

 

<math>\sin (A - B) = \sin A \cos B - \cos A \sin B \,</math>

<math>\cos (A - B) = \cos A \cos B + \sin A \sin B \,</math>

<math>\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} = \frac{\cot B - \cot A}{1 + \cot A \cot B} \,</math>

Double-angle identities

<math>\sin 2A = 2 \sin A \cos A \,</math>

<math>\cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A -1 = 1-2 \sin^2 A = {1 - \tan^2 A \over 1 + \tan^2 A} \,</math>

<math>\tan 2A = {2 \tan A \over 1 - \tan^2 A} = {2 \cot A \over \cot^2 A - 1} = {2 \over \cot A - \tan A} \,</math>

Half-angle identities

Note that <math>\pm</math> are correct, it means it may be either one, depending on the value of A/2.

<math>\sin \frac{A}{2} = \pm \sqrt{\frac{1-\cos A}{2}} \,</math>

<math>\cos \frac{A}{2} = \pm \sqrt{\frac{1+\cos A}{2}} \,</math>

<math>\tan \frac{A}{2} = \pm \sqrt{\frac{1-\cos A}{1+\cos A}} = \frac {\sin A}{1+\cos A} = \frac {1-\cos A}{\sin A} \,</math>

Law of tangents

The law of tangents:

<math>\frac{a+b}{a-b} = \frac{\tan[\frac{1}{2}(A+B)]}{\tan[\frac{1}{2}(A-B)]}</math>