PlanetMath: double angle identity

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double angle identity

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The double angle identities are

$\displaystyle \sin(2x)$	$\displaystyle =$	$\displaystyle 2\sin{x}\cos{x}$	(1)
$\displaystyle \cos(2x)$	$\displaystyle =$	$\displaystyle \cos^2{x}-\sin^2{x} = 2\cos^2{x}-1 = 1-2\sin^2{x}$	(2)
$\displaystyle \tan(2x)$	$\displaystyle =$	$\displaystyle \frac{2\tan{x}}{1-\tan^2{x}}$	(3)

These are all derived from their respective trigonometric addition formulas. For example,

$\displaystyle \sin(2x)$	$\displaystyle =$	$\displaystyle \sin(x+x)$
	$\displaystyle =$	$\displaystyle \sin{x}\cos{x}+\cos{x}\sin{x}$
	$\displaystyle =$	$\displaystyle 2\sin{x}\cos{x}$

The formula for cosine follows similarly, and the formula tangent is derived by taking the ratio of sine to cosine, as always.

The double angle identities can also be derived from the de Moivre identity.

"double angle identity" is owned by Wkbj79. [ full author list (2) | owner history (1) ]

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Other names:

double-angle identity, double angle formula, double-angle formula, double angle formulae, double-angle formulae

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Cross-references: de Moivre identity, sine, ratio, tangent, cosine, trigonometric addition formulas
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This is version 12 of double angle identity, born on 2002-01-30, modified 2007-06-25.
Object id is 1623, canonical name is DoubleAngleIdentity.
Accessed 26537 times total.

Classification:

AMS MSC:	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)

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