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List of trigonometric identities

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fuckFundamental IdentitiesEdit
"arc" Identities
  • \displaystyle\arctan\theta=\tan^{-1}\theta
  • \displaystyle\arcsin\theta=\sin^{-1}\theta
  • \displaystyle\arccos\theta=\cos^{-1}\theta
Quotient and reciprocal identities
  • \displaystyle\tan\theta=\frac{\sin\theta}{\cos\theta}
  • \displaystyle\cot\theta=\frac{\cos\theta}{\sin\theta}= \frac{\csc\theta}{\sec\theta}= \frac{1}{\tan\theta}
  • \displaystyle\sec\theta=\frac{1}{\cos\theta}
  • \displaystyle\csc\theta=\frac{1}{\sin\theta}
Cofunction Function identities
  • \displaystyle\sin\theta = \cos(\frac{\pi}{2} - \theta)
  • \displaystyle\cos\theta = \sin(\frac{\pi}{2} - \theta)
Even/Odd Functions
  • \displaystyle\cos(-\theta) = \cos(\theta)
  • \displaystyle\sin(-\theta) = -\sin(\theta)
Pythagorean identities
  • \displaystyle\sin^2\theta+\cos^2\theta=1
  • \displaystyle\tan^2\theta+1=\sec^2\theta
  • \displaystyle1+\cot^2\theta=\csc^2\theta
Angle sum and difference identities
  • \displaystyle\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha
  • \displaystyle\sin(\alpha-\beta)=\sin\alpha\cos\beta-\sin\beta\cos\alpha
  • \displaystyle\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
  • \displaystyle\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta
  • \displaystyle\tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
  • \displaystyle\tan(\alpha-\beta) = \frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}
Double-angle identities
  • \displaystyle\sin2\theta=2\sin\theta\cos\theta
  • \displaystyle\cos2\theta=\cos^2\theta-\sin^2\theta = 2\cos^2\theta-1 = 1-2\sin^2\theta
  • \displaystyle\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}
Half-angle identities
  • \displaystyle\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}
  • \displaystyle\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}
  • \displaystyle\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta}

Reduction formulas

  • \displaystyle\sin^2\theta=\frac{1-\cos2\theta}{2}
  • \displaystyle\cos^2\theta=\frac{1+\cos2\theta}{2}
  • \displaystyle\tan^2\theta=\frac{1-\cos2\theta}{1+\cos2\theta} = \frac{\sin2\theta}{1+\cos2\theta} = \frac{1-\cos2\theta}{\sin2\theta}

Product-to-sum identities

  • \displaystyle\cos a\cos b = \frac{1}{2} [\cos (a-b) + \cos (a+b)]
  • \displaystyle\sin a\sin b = \frac{1}{2} [\cos (a-b) - \cos (a+b)]
  • \displaystyle\sin a\cos b = \frac{1}{2} [\sin (a+b) + \sin (a-b)]
  • \displaystyle\cos a\sin b = \frac{1}{2} [\sin (a+b) - \cos (a-b)]

Sum-to-product identities

  • \displaystyle\cos a + \cos b = 2\cos \left(\frac{a+b}{2} \right) \cos \left(\frac{a-b}{2} \right)
  • \displaystyle\cos a - \cos b = -2\sin \left(\frac{a+b}{2} \right) \sin \left(\frac{a-b}{2} \right)
  • \displaystyle\sin a + \sin b = 2\sin \left(\frac{a+b}{2} \right) \cos \left(\frac{a-b}{2} \right)
  • \displaystyle\sin a - \sin b = 2\sin \left(\frac{a-b}{2} \right) \cos \left(\frac{a+b}{2} \right)

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