Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
[ "article:topic", "showtoc:no" ]
Mathematics LibreTexts

Trigonometric Identities

  • Page ID
    18803
  •  

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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

    \(\sin(α + β) = \sin(α) \cos(β) + \cos(α) \sin(β)\)

    \(\sin(α - β) = \sin(α) \cos(β) - \cos(α) \sin(β)\)

    \(\cos(α + β) = \cos(α) \cos(β) - \sin(α) \sin(β)\)

    \(\cos(α - β) = \cos(α) \cos(β) + \sin(α) \sin(β)\)