Trigonometric Identities
- Page ID
- 18803
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Pythagorean Identities
\(\cos^2 x + \sin^2 x = 1\)
\(\sec^2 x - \tan^2 x = 1\)
Double-Angle Identities
\(\sin 2x = 2 \sin x \cos x\)
\(\cos 2x = \cos^2 x - \sin^2 x = 1 - 2 \sin^2 x = 2 \cos^2 x - 1\)
Half-Angle Identities
\(\cos^2 x = \dfrac{1+ \cos 2x}{2}\)
\(\sin^2 x = \dfrac{1- \cos 2x}{2}\)
Angle Sum and Difference Identities
\(\sin(α + β) = \sin(α) \cos(β) + \cos(α) \sin(β)\)
\(\sin(α - β) = \sin(α) \cos(β) - \cos(α) \sin(β)\)
\(\cos(α + β) = \cos(α) \cos(β) - \sin(α) \sin(β)\)
\(\cos(α - β) = \cos(α) \cos(β) + \sin(α) \sin(β)\)
Angle Reflections and Shifts
\(\sin (-x) = -\sin x\)
\(\cos(-x) = \cos x\)
\(\tan (-x) = -\tan x\)
\(\sin\left(x \pm \frac{\pi}{2}\right) = \pm \cos x\)
\(\cos\left(x \pm \frac{\pi}{2}\right) = \mp \sin x\)
Angle Supplement Identities
\(\sin(\pi - x) = \sin x\)
\(\cos(\pi - x) = -\cos x\)
\(\tan(\pi - x) = -\tan x\)
Periodicity Identities
\(\sin(x+2\pi) = \sin x\)
\(\cos(x+2\pi) = \cos x\)
\(\tan(x+\pi) = \tan x\)