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Trigonometric identities

$\displaystyle \sin(-\alpha)$ $\displaystyle = -\sin\alpha$ (A.25)
$\displaystyle \cos(-\alpha)$ $\displaystyle =+ \cos\alpha$ (A.26)
$\displaystyle \tan(-\alpha)$ $\displaystyle = -\tan\alpha$ (A.27)
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =1$ (A.28)
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =\sin\alpha\,\cos\beta\pm \cos\alpha\,\sin\beta$ (A.29)
$\displaystyle \cos(\alpha\pm \beta)$ $\displaystyle =\cos\alpha\,\cos\beta\mp \sin\alpha\,\sin\beta$ (A.30)
$\displaystyle \tan(\alpha\pm \beta)$ $\displaystyle = =\frac{\tan\alpha\pm \tan\beta}{1\mp \tan\alpha\,\tan\beta}$ (A.31)

$\displaystyle \sin\alpha+\sin\beta$ $\displaystyle =2\,\sin\left(\frac{\alpha+\beta}{2}\right)\,\cos\left(\frac{\alpha-\beta}{2}\right)$ (A.32)
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =2\,\cos\left(\frac{\alpha+\beta}{2}\right)\,\sin\left(\frac{\alpha-\beta}{2}\right)$ (A.33)
$\displaystyle \cos\alpha+\cos\beta$ $\displaystyle =2\,\cos\left(\frac{\alpha+\beta}{2}\right)\,\cos\left(\frac{\alpha-\beta}{2}\right)$ (A.34)
$\displaystyle \cos\alpha-\cos\beta$ $\displaystyle =-2\,\sin\left(\frac{\alpha+\beta}{2}\right)\,\sin\left(\frac{\alpha-\beta}{2}\right)$ (A.35)
$\displaystyle \sin\alpha\,\sin\beta$ $\displaystyle =\frac{1}{2}\left[\cos(\alpha-\beta)-\cos(\alpha+\beta)\right]$ (A.36)
$\displaystyle \cos\alpha\,\cos\beta$ $\displaystyle =\frac{1}{2}\left[\cos(\alpha-\beta)+\cos(\alpha+\beta)\right]$ (A.37)
$\displaystyle \sin\alpha\,\cos\beta$ $\displaystyle =\frac{1}{2}\left[\sin(\alpha-\beta)+\sin(\alpha+\beta)\right]$ (A.38)
$\displaystyle \sin(\alpha/2)$ $\displaystyle =\pm\left(\frac{1-\cos\alpha}{2}\right)^{1/2}$ (A.39)
$\displaystyle \cos(\alpha/2)$ $\displaystyle = \pm \left(\frac{1+\cos\alpha}{2}\right)^{1/2}$ (A.40)
$\displaystyle \tan(\alpha/2)$ $\displaystyle = \pm\left(\frac{1-\cos\alpha}{1+\cos\alpha}\right)^{1/2} = \frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}$ (A.41)
$\displaystyle \sin(2\alpha)$ $\displaystyle = 2\,\sin\alpha\,\cos\alpha$ (A.42)
$\displaystyle \cos(2\alpha)$ $\displaystyle =\cos^2\alpha-\sin^2\alpha = 2\,\cos^2\alpha-1=1-2\,\sin^2\alpha$ (A.43)
$\displaystyle \sin(3\alpha)$ $\displaystyle =-4\,\sin^3\alpha+3\,\sin\alpha$ (A.44)
$\displaystyle \cos(3\alpha)$ $\displaystyle =4\,\cos^3\alpha-3\,\cos\alpha$ (A.45)
$\displaystyle \sin(4\alpha)$ $\displaystyle = (-8\,\sin^3\alpha+4\,\sin\alpha)\,\cos\alpha$ (A.46)
$\displaystyle \cos(4\alpha)$ $\displaystyle =8\,\cos^4\alpha-8\,\cos^2\alpha+1$ (A.47)
$\displaystyle \sin^2\alpha$ $\displaystyle =\frac{1}{2}\,(1-\cos 2\alpha)$ (A.48)
$\displaystyle \cos^2\alpha$ $\displaystyle =\frac{1}{2}\,(1+\cos 2\alpha)$ (A.49)
$\displaystyle \sin^3\alpha$ $\displaystyle = \frac{1}{4}\,(3\,\sin\alpha-\sin 3\alpha)$ (A.50)
$\displaystyle \cos^3\alpha$ $\displaystyle =\frac{1}{4}\,(3\,\cos\alpha+\cos 3\alpha)$ (A.51)
$\displaystyle \sin^4\alpha$ $\displaystyle = \frac{1}{8}\,(3-4\,\cos 2\alpha+\cos 4\alpha)$ (A.52)
$\displaystyle \cos^4\alpha$ $\displaystyle = \frac{1}{8}\,(3+4\,\cos 2\alpha+\cos 4\alpha)$ (A.53)

$\displaystyle \sinh(-\alpha)$ $\displaystyle = -\sinh\alpha$ (A.54)
$\displaystyle \cosh(-\alpha)$ $\displaystyle =+ \cosh\alpha$ (A.55)
$\displaystyle \tanh(-\alpha)$ $\displaystyle = -\tanh\alpha$ (A.56)
$\displaystyle \cosh^2\alpha-\sinh^2\alpha$ $\displaystyle =1$ (A.57)
$\displaystyle \sinh(\alpha\pm\beta)$ $\displaystyle =\sinh\alpha\,\cosh\beta\pm \cosh\alpha\,\sinh\beta$ (A.58)
$\displaystyle \cosh(\alpha\pm \beta)$ $\displaystyle =\cosh\alpha\,\cosh\beta\pm \sinh\alpha\,\sinh\beta$ (A.59)
$\displaystyle \tanh(\alpha\pm \beta)$ $\displaystyle = \frac{\tanh\alpha\pm\tanh\beta}{1\pm\tanh\alpha\,\tanh\beta}$ (A.60)
$\displaystyle \sinh\alpha+\sinh\beta$ $\displaystyle =2\,\sinh\left(\frac{\alpha+\beta}{2}\right)\,\cosh\left(\frac{\alpha-\beta}{2}\right)$ (A.61)
$\displaystyle \sinh\alpha-\sinh\beta$ $\displaystyle =2\,\cosh\left(\frac{\alpha+\beta}{2}\right)\,\sinh\left(\frac{\alpha-\beta}{2}\right)$ (A.62)
$\displaystyle \cosh\alpha+\cosh\beta$ $\displaystyle =2\,\cosh\left(\frac{\alpha+\beta}{2}\right)\,\cosh\left(\frac{\alpha-\beta}{2}\right)$ (A.63)
$\displaystyle \cosh\alpha-\cosh\beta$ $\displaystyle =2\,\sinh\left(\frac{\alpha+\beta}{2}\right)\,\sinh\left(\frac{\alpha-\beta}{2}\right)$ (A.64)
$\displaystyle \sinh\alpha\,\sinh\beta$ $\displaystyle =\frac{1}{2}\left[\cosh(\alpha+\beta)-\cosh(\alpha-\beta)\right]$ (A.65)
$\displaystyle \cosh\alpha\,\cosh\beta$ $\displaystyle =\frac{1}{2}\left[\cosh(\alpha+\beta)+\cosh(\alpha-\beta)\right]$ (A.66)
$\displaystyle \sinh\alpha\,\cosh\beta$ $\displaystyle =\frac{1}{2}\left[\sinh(\alpha+\beta)+\sinh(\alpha-\beta)\right]$ (A.67)
$\displaystyle \sinh(\alpha/2)$ $\displaystyle = \left(\frac{\cosh\alpha-1}{2}\right)^{1/2}$ (A.68)
$\displaystyle \cosh(\alpha/2)$ $\displaystyle =\left(\frac{\cosh\alpha+1}{2}\right)^{1/2}$ (A.69)
$\displaystyle \tanh(\alpha/2)$ $\displaystyle = \left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)^{1/2}=\frac{\cosh\alpha-1}{\sinh\alpha}= \frac{\sinh\alpha}{\cosh\alpha+1}$ (A.70)
$\displaystyle \sinh(2\alpha)$ $\displaystyle = 2\,\sinh\alpha\,\cosh\alpha$ (A.71)
$\displaystyle \cosh(2\alpha)$ $\displaystyle =\cosh^2\alpha+\sinh^2\alpha = 2\,\cosh^2\alpha-1=2\,\sinh^2\alpha+1$ (A.72)


next up previous
Next: Vector identities Up: Useful mathematics Previous: Series expansions
Richard Fitzpatrick 2016-03-31