Summary of Trigonometric Identities

31 Aug 2022

The Unit Circle and Common Angles

Complementary, Supplementary, and Opposite Angles

\[\begin{align} &\cos(\pi/2 - \theta) = \sin(\theta) &\qquad& \sin(\pi/2 - \theta) = \cos(\theta) &\qquad& \tan(\pi/2 - \theta) = \cot(\theta) \\[10pt] &\cos(\pi - \theta) = -\cos(\theta) &\qquad& \sin(\pi - \theta) = \sin(\theta) &\qquad& \tan(\pi - \theta) = -\tan(\theta) \\[10pt] &\cos(- \theta) = \cos(\theta) &\qquad& \sin(- \theta) = -\sin(\theta) &\qquad& \tan(- \theta) = -\tan(\theta) \end{align}\]

\[\begin{align} &\sec(\pi/2 - \theta) = -\sec(\theta) &\qquad& \csc(\pi/2 - \theta) = \csc(\theta) &\qquad& \cot(\pi/2 - \theta) = \tan(\theta) \\[10pt] &\sec(\pi - \theta) = \csc(\theta) &\qquad& \csc(\pi - \theta) = \sec(\theta) &\qquad& \cot(\pi - \theta) = -\cot(\theta) \\[10pt] &\sec(- \theta) = \sec(\theta) &\qquad& \csc(- \theta) = -\csc(\theta) &\qquad& \cot(- \theta) = -\cot(\theta) \end{align}\]

Pythagorean Identities

\[\sin^2 \theta + \cos^2 \theta = 1 \qquad \tan^2 \theta + 1 = \sec^2 \theta \qquad 1 + \cot^2 \theta = \csc^2 \theta\]

Sum and Difference Identities

\[\begin{align} &\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta) &\qquad\qquad& \sec(\alpha \pm \beta) = \frac{\sec(\alpha)\sec(\beta)\csc(\alpha)\csc(\beta)}{\csc(\alpha)\csc(\beta) \mp \sec(\alpha)\sec(\beta)} \\[15pt] &\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta) &\qquad\qquad& \csc(\alpha \pm \beta) = \frac{\sec(\alpha)\sec(\beta)\csc(\alpha)\csc(\beta)}{\sec(\alpha)\csc(\beta) \pm \csc(\alpha)\sec(\beta)} \\[15pt] &\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)} &\qquad\qquad& \cot(\alpha \pm \beta) = \frac{\cot(\alpha) \cot(\beta) \mp 1}{\cot(\beta) \pm \cot(\alpha)} \end{align}\]

Double Angle Identities

\[\begin{align} &\cos(2\theta) = \cos^2 \theta - \sin^2 \theta \quad = \quad 1 - 2\sin^2 \theta \quad = \quad 2\cos^2 \theta - 1 \\[15pt] &\sin(2\theta) = 2 \sin \theta \cos \theta \\[15pt] &\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta} \quad = \quad \frac{2}{\cot \theta - \tan \theta} \\[15pt] &\sec(2 \theta) = \frac{\sec^2 \theta \csc^2 \theta}{\csc^2 \theta - \sec^2 \theta} \quad = \quad \frac{1 + \tan^2 \theta}{1 - \tan^2 \theta} \\[15pt] &\csc(2 \theta) = \frac{1}{2}\sec \theta \csc \theta \quad = \quad \frac{1}{2} \left ( \tan \theta + \cot \theta \right ) \\[15pt] &\cot(2 \theta) = \frac{\cot^2 \theta - 1}{2\cot \theta} \quad = \quad \frac{1}{2} (\cot \theta - \tan \theta) \end{align}\]

Power Reduction Identities

\[\cos^2 \theta = \frac{1}{2}(1 + \cos(2 \theta)) \qquad\qquad \sin^2 \theta = \frac{1}{2}(1 - \cos(2 \theta)) \qquad\qquad \tan^2 \theta = \frac{1 - \cos(2 \theta)}{1 + \cos(2 \theta)}\]

Half Angle Identities

\[\cos (\theta/2) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \qquad\qquad \sin (\theta/2) = \pm \sqrt{\frac{1 - \cos \theta}{2}}\] \[\tan (\theta/2) = \pm \sqrt{ \frac{1 - \cos \theta}{1 + \cos \theta} } \quad=\quad \pm \frac{\sin \theta}{1 + \cos \theta} \quad=\quad \pm \frac{1 - \cos \theta}{\sin \theta}\]

Product-to-Sum Identities

\[\begin{align} &\cos \alpha \cos \beta = \frac{1}{2} (\cos(\alpha - \beta) + \cos(\alpha + \beta)) \qquad\qquad &\sin \alpha \cos \beta = \frac{1}{2} (\sin(\alpha + \beta) + \sin(\alpha - \beta)) \\[15pt] &\sin \alpha \sin \beta = \frac{1}{2} (\cos(\alpha - \beta) - \cos(\alpha + \beta)) \qquad\qquad &\cos \alpha \sin \beta = \frac{1}{2} (\sin(\alpha + \beta) - \sin(\alpha - \beta)) \end{align}\]

Sum-to-Product Identities

\[\begin{align} &\cos \alpha + \cos \beta = 2 \cos \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) &\qquad &\sin \alpha + \sin \beta = 2 \sin \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right )\\[15pt] &\cos \alpha - \cos \beta = 2 \sin \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) &\qquad &\sin \alpha - \sin \beta = 2 \cos \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \end{align}\]

Euler’s Identity

\[e^{i\theta} = \cos \theta + i \sin \theta \qquad\qquad \cos \theta = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) \qquad\qquad \sin \theta = \frac{1}{2i} (e^{i\theta} - e^{-i\theta})\]

Hyperbolic Trigonometric Functions

Exponential Definitions

\[\begin{align} \sinh x &= \frac{e^{x} - e^{-x}}{2} = \frac{e^{2x} - 1}{2e^{x}} = \frac{1 - e^{-2x}}{2e^{-x}} \\[15pt] \cosh x &= \frac{e^{x} + e^{-x}}{2} = \frac{e^{2x} + 1}{2e^{x}} = \frac{1 + e^{-2x}}{2e^{-x}} \\[15pt] \tanh x &= \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} = \frac{e^{2x} - 1}{e^{2x} + 1} \\[15pt] \csch x &= \frac{1}{\sinh x} \\[10pt] \sech x &= \frac{1}{\cosh x} \\[10pt] \coth x &= \frac{1}{\tanh x} = \frac{\cosh x}{\sinh x} \end{align}\]

Relationship to Standard Trigonometric Functions

\[\begin{align} &\sinh x = -i \sin (i x) &\qquad\qquad& \sin x = -i \sinh(ix) \\[10pt] &\cosh x = \cos (i x) &\qquad\qquad& \cos x = \cosh(ix) \\[10pt] &\tanh x = -i \tan (i x) &\qquad\qquad& \tan x = -i \tanh(ix) \\[10pt] &\sech x = \sec (i x) &\qquad\qquad& \sec x = \sech(ix) \\[10pt] &\csch x = i \csc (i x) &\qquad\qquad& \csc x = i \csch(ix) \\[10pt] &\coth x = i \cot (i x) &\qquad\qquad& \cot x = i \coth(ix) \end{align}\]

Inverse Functions

\[\begin{align} &\arcsinh(x) = \ln \left ( x + \sqrt{x^2 + 1} \right ) && \text{for all } x \\[10pt] &\arccosh(x) = \ln \left ( x + \sqrt{x^2 - 1} \right ) && x \geq 1 \\[10pt] &\arctanh(x) = \frac{1}{2} \ln \left ( \frac{1+x}{1-x} \right ) && \abs{x} < 1 \\[10pt] &\arcsech(x) = \ln \left ( \frac{1}{x} + \sqrt{\frac{1}{x^2} - 1} \right ) && x \neq 0 \\[10pt] &\arccsch(x) = \ln \left ( \frac{1}{x} + \sqrt{\frac{1}{x^2} + 1} \right ) && 0 < x \leq 1 \\[10pt] &\arccoth(x) = \frac{1}{2} \ln \left ( \frac{x+1}{x-1} \right ) && \abs{x} > 1 \end{align}\]

Trigonometry Series