14.3.3: Trigonometric Identities
- Page ID
- 61034
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
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
\(\sin (-x) = -\sin x\)
\(\cos(-x) = \cos x\)
\(\sin\left(x \pm \frac{\pi}{2}\right) = \pm \cos x\)
\(\cos\left(x \pm \frac{\pi}{2}\right) = \mp \sin x\)