Trigonometric Addition 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

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

Also defined as

Some sources include the Trigonometric Subtraction Formulas in with the Trigonometric Addition Formulas.

Other sources lump the Trigonometric Addition Formulas and Trigonometric Subtraction Formulas together and call them all Compound Angle Formulas.

Also known as

The trigonometric addition formulas are also known as the trigonometric addition formulae.

They are also known as:

• the addition formulas or addition formulae

• Ptolemy's formulas or Ptolemy's formulae, for Claudius Ptolemy.

• Simpson's formulas or Simpson's formulae, for Thomas Simpson, although this name has been given to various sets of trigonometric identities, which can cause confusion.

Some sources use the term trigonometric addition formulas as a synonym for compound angle formulas.

Also see

• Trigonometric Subtraction Formulas

• Addition Formulas for Hyperbolic Functions
• Subtraction Formulas for Hyperbolic Functions

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): addition formula: 1.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition formulae
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition formulae