# Compound Angle Formulas

## Theorem

### Sine of Sum

$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$

### Cosine of Sum

$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$

### Tangent of Sum

$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$

## Hyperbolic Functions

### Hyperbolic Sine of Sum

$\map \sinh {a + b} = \sinh a \cosh b + \cosh a \sinh b$

### Hyperbolic Cosine of Sum

$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$

### Hyperbolic Tangent of Sum

$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$

### Hyperbolic Cotangent of Sum

$\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$

## Also known as

The compound angle formulas are also known as the addition formulas.

They are also known as Ptolemy's formulas for Claudius Ptolemy.

The form formulae can also be seen in the literature.