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.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): addition formula
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Ptolemy's formulae
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Ptolemy's formulae
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): addition formula
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): compound angle formulae (in trigonometry)