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Trigonometric Identities

  • Page ID
    58646
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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

    \(\sin(x+2\pi) = \sin x\)

    \(\cos(x+2\pi) = \cos x\)

    \(\tan(x+\pi) = \tan x\)

     


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