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Next: Electromagnetic Theory Up: Useful Mathematics Previous: Series Expansions


Trigonometric Identities

$\displaystyle \sin(-\alpha)$ $\displaystyle = -\sin\alpha$ (1236)
$\displaystyle \cos(-\alpha)$ $\displaystyle =+ \cos\alpha$ (1237)
$\displaystyle \tan(-\alpha)$ $\displaystyle = -\tan\alpha$ (1238)
$\displaystyle \sin(\alpha\pm\pi/2)$ $\displaystyle =\pm\cos\alpha$ (1239)
$\displaystyle \cos(\alpha\pm\pi/2)$ $\displaystyle =\mp \sin\alpha$ (1240)
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =1$ (1241)
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =\sin\alpha \cos\beta\pm \cos\alpha \sin\beta$ (1242)
$\displaystyle \cos(\alpha\pm \beta)$ $\displaystyle =\cos\alpha \cos\beta\mp \sin\alpha \sin\beta$ (1243)
$\displaystyle \tan(\alpha\pm \beta)$ $\displaystyle = \frac{\tan\alpha\pm \tan\beta}{1\mp \tan\alpha\,\tan\beta}$ (1244)
$\displaystyle \sin\alpha+\sin\beta$ $\displaystyle =2\,\sin\left(\frac{\alpha+\beta}{2}\right)\,\cos\left(\frac{\alpha-\beta}{2}\right)$ (1245)
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =2\,\cos\left(\frac{\alpha+\beta}{2}\right)\,\sin\left(\frac{\alpha-\beta}{2}\right)$ (1246)
$\displaystyle \cos\alpha+\cos\beta$ $\displaystyle =2\,\cos\left(\frac{\alpha+\beta}{2}\right)\,\cos\left(\frac{\alpha-\beta}{2}\right)$ (1247)
$\displaystyle \cos\alpha-\cos\beta$ $\displaystyle =-2\,\sin\left(\frac{\alpha+\beta}{2}\right)\,\sin\left(\frac{\alpha-\beta}{2}\right)$ (1248)
$\displaystyle \sin\alpha \sin\beta$ $\displaystyle =\frac{1}{2}\left[\cos(\alpha-\beta)-\cos(\alpha+\beta)\right]$ (1249)
$\displaystyle \cos\alpha\,\cos\beta$ $\displaystyle =\frac{1}{2}\left[\cos(\alpha-\beta)+\cos(\alpha+\beta)\right]$ (1250)
$\displaystyle \sin\alpha \cos\beta$ $\displaystyle =\frac{1}{2}\left[\sin(\alpha-\beta)+\sin(\alpha+\beta)\right]$ (1251)
$\displaystyle \sin(\alpha/2)$ $\displaystyle = \pm\left(\frac{1-\cos\alpha}{2}\right)^{1/2}$ (1252)
$\displaystyle \cos(\alpha/2)$ $\displaystyle = \pm \left(\frac{1+\cos\alpha}{2}\right)^{1/2}$ (1253)
$\displaystyle \tan(\alpha/2)$ $\displaystyle = \pm\left(\frac{1-\cos\alpha}{1+\cos\alpha}\right)^{1/2} = \frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}$ (1254)
$\displaystyle \sin(2\alpha)$ $\displaystyle = 2 \sin\alpha \cos\alpha$ (1255)
$\displaystyle \cos(2\alpha)$ $\displaystyle =\cos^2\alpha-\sin^2\alpha = 2 \cos^2\alpha-1$    
  $\displaystyle    =1-2 \sin^2\alpha$ (1256)

$\displaystyle \sin(3\alpha)$ $\displaystyle =-4 \sin^3\alpha+3 \sin\alpha$ (1257)
$\displaystyle \cos(3\alpha)$ $\displaystyle =4\,\cos^3\alpha-3\,\cos\alpha$ (1258)
$\displaystyle \sin(4\alpha)$ $\displaystyle = (-8 \sin^3\alpha+4 \sin\alpha) \cos\alpha$ (1259)
$\displaystyle \cos(4\alpha)$ $\displaystyle =8\,\cos^4\alpha-8\,\cos^2\alpha+1$ (1260)
$\displaystyle \sin^2\alpha$ $\displaystyle =\frac{1}{2} (1-\cos 2\alpha)$ (1261)
$\displaystyle \cos^2\alpha$ $\displaystyle =\frac{1}{2} (1+\cos 2\alpha)$ (1262)
$\displaystyle \sin^3\alpha$ $\displaystyle = \frac{1}{4} (3 \sin\alpha-\sin 3\alpha)$ (1263)
$\displaystyle \cos^3\alpha$ $\displaystyle =\frac{1}{4} (3 \cos\alpha+\cos 3\alpha)$ (1264)
$\displaystyle \sin^4\alpha$ $\displaystyle = \frac{1}{8} (3-4 \cos 2\alpha+\cos 4\alpha)$ (1265)
$\displaystyle \cos^4\alpha$ $\displaystyle = \frac{1}{8} (3+4 \cos 2\alpha+\cos 4\alpha)$ (1266)
$\displaystyle \sinh(-\alpha)$ $\displaystyle = -\sinh\alpha$ (1267)
$\displaystyle \cosh(-\alpha)$ $\displaystyle =+ \cosh\alpha$ (1268)
$\displaystyle \tanh(-\alpha)$ $\displaystyle = -\tanh\alpha$ (1269)
$\displaystyle \cosh^2\alpha-\sinh^2\alpha$ $\displaystyle =1$ (1270)
$\displaystyle \sinh(\alpha\pm\beta)$ $\displaystyle =\sinh\alpha \cosh\beta\pm \cosh\alpha \sinh\beta$ (1271)
$\displaystyle \cosh(\alpha\pm \beta)$ $\displaystyle =\cosh\alpha \cosh\beta\pm \sinh\alpha \sinh\beta$ (1272)
$\displaystyle \tanh(\alpha\pm \beta)$ $\displaystyle = \frac{\tanh\alpha\pm\tanh\beta}{1\pm\tanh\alpha\,\tanh\beta}$ (1273)
$\displaystyle \sinh\alpha+\sinh\beta$ $\displaystyle =2\,\sinh\left(\frac{\alpha+\beta}{2}\right)\,\cosh\left(\frac{\alpha-\beta}{2}\right)$ (1274)
$\displaystyle \sinh\alpha-\sinh\beta$ $\displaystyle =2\,\cosh\left(\frac{\alpha+\beta}{2}\right)\,\sinh\left(\frac{\alpha-\beta}{2}\right)$ (1275)
$\displaystyle \cosh\alpha+\cosh\beta$ $\displaystyle =2\,\cosh\left(\frac{\alpha+\beta}{2}\right)\,\cosh\left(\frac{\alpha-\beta}{2}\right)$ (1276)
$\displaystyle \cosh\alpha-\cosh\beta$ $\displaystyle =2\,\sinh\left(\frac{\alpha+\beta}{2}\right)\,\sinh\left(\frac{\alpha-\beta}{2}\right)$ (1277)
$\displaystyle \sinh\alpha \sinh\beta$ $\displaystyle =\frac{1}{2}\left[\cosh(\alpha+\beta)-\cosh(\alpha-\beta)\right]$ (1278)
$\displaystyle \cosh\alpha \cosh\beta$ $\displaystyle =\frac{1}{2}\left[\cosh(\alpha+\beta)+\cosh(\alpha-\beta)\right]$ (1279)
$\displaystyle \sinh\alpha \cosh\beta$ $\displaystyle =\frac{1}{2}\left[\sinh(\alpha+\beta)+\sinh(\alpha-\beta)\right]$ (1280)

$\displaystyle \sinh(\alpha/2)$ $\displaystyle =\pm \left(\frac{\cosh\alpha-1}{2}\right)^{1/2}$ (1281)
$\displaystyle \cosh(\alpha/2)$ $\displaystyle =\left(\frac{\cosh\alpha+1}{2}\right)^{1/2}$ (1282)
$\displaystyle \tanh(\alpha/2)$ $\displaystyle =\pm \left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)^{1/2}=\frac{\cosh\alpha-1}{\sinh\alpha}$    
  $\displaystyle ~~~~= \frac{\sinh\alpha}{\cosh\alpha+1}$ (1283)
$\displaystyle \sinh(2\alpha)$ $\displaystyle = 2 \sinh\alpha \cosh\alpha$ (1284)
$\displaystyle \cosh(2\alpha)$ $\displaystyle =\cosh^2\alpha+\sinh^2\alpha = 2 \cosh^2\alpha-1$    
  $\displaystyle    =2 \sinh^2\alpha+1$ (1285)

$\displaystyle \cos\theta$ $\displaystyle = \frac{1}{2}\left({\rm e}^{ {\rm i} \theta}+{\rm e}^{-{\rm i} \theta}\right)$ (1286)
$\displaystyle \sin\theta$ $\displaystyle = \frac{1}{2 {\rm i}}\left({\rm e}^{ {\rm i} \theta}-{\rm e}^{-{\rm i} \theta}\right)$ (1287)
$\displaystyle \cosh\theta$ $\displaystyle = \frac{1}{2}\left({\rm e}^{ \theta}+{\rm e}^{-\theta}\right)$ (1288)
$\displaystyle \sinh\theta$ $\displaystyle = \frac{1}{2}\left({\rm e}^{ \theta}-{\rm e}^{-\theta}\right)$ (1289)
$\displaystyle \cos({\rm i} \theta)$ $\displaystyle = \cosh\theta$ (1290)
$\displaystyle \sin({\rm i} \theta)$ $\displaystyle = {\rm i} \sinh\theta$ (1291)
$\displaystyle \cosh({\rm i} \theta)$ $\displaystyle = \cos\theta$ (1292)
$\displaystyle \sinh({\rm i} \theta)$ $\displaystyle = {\rm i} \sin\theta$ (1293)

Source: Spiegel and Liu 1999.


next up previous
Next: Electromagnetic Theory Up: Useful Mathematics Previous: Series Expansions
Richard Fitzpatrick 2013-04-08