Angle Addition Formulae

Level 1 - Math I (Physics) topic page in Trigonometry.

Angle Addition Formulas

These formulas express trigonometric functions of sums and differences of angles in terms of functions of the individual angles.

Sine sum

\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]

Sine difference

\[\sin(A - B) = \sin A \cos B - \cos A \sin B\]

Cosine sum

\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]

Cosine difference

\[\cos(A - B) = \cos A \cos B + \sin A \sin B\]

Tangent sum

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

Tangent difference

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

Setting \(B = A\) in the sum formulas gives the double angle formulas:

Sine double

\[\sin(2A) = 2\sin A \cos A\]

Cosine double

\[\cos(2A) = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A\]

Tangent double

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

Derived from the double angle formulas:

Sine half

\[\sin\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{2}}\]

Cosine half

\[\cos\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 + \cos A}{2}}\]

Tangent half

\[\tan\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A}\]

The \(\pm\) sign depends on the quadrant in which the angle lies.