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Mathematics LibreTexts

Fundamental Trigonometric Identities

 

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"arc" Identities

\[\arctan\theta=\tan^{-1}\theta\]

\[\arcsin\theta=\sin^{-1}\theta\]

\[\arccos\theta=\cos^{-1}\theta\]

Quotient and reciprocal identities

\[\tan\theta=\dfrac{\sin\theta}{\cos\theta}\]

\[\cot\theta=\dfrac{\cos\theta}{\sin\theta}= \dfrac{\csc\theta}{\sec\theta}= \dfrac{1}{\tan\theta}\]

\[\sec\theta=\dfrac{1}{\cos\theta}\]

\[\csc\theta=\dfrac{1}{\sin\theta}\]

Cofunction Function identities

\[\sin\theta = \cos(\dfrac{\pi}{2} - \theta)\]

\[\cos\theta = \sin(\dfrac{\pi}{2} - \theta)\]

Even/Odd Functions

\[\cos(-\theta) = \cos(\theta)\]

\[\sin(-\theta) = -\sin(\theta)\]

Pythagorean identities

\[\sin^2\theta+\cos^2\theta=1\]

\[\tan^2\theta+1=\sec^2\theta\]

\[1+\cot^2\theta=\csc^2\theta\]

Angle sum and difference identities

\[\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha\]

\[\sin(\alpha-\beta)=\sin\alpha\cos\beta-\sin\beta\cos\alpha\]

\[\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\]

\[\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\]

\[\tan(\alpha+\beta) = \dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\]

\[\tan(\alpha-\beta) = \dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}\]

Double-angle identities

\[\sin2\theta=2\sin\theta\cos\theta\]

\[\cos2\theta=\cos^2\theta-\sin^2\theta = 2\cos^2\theta-1 = 1-2\sin^2\theta\]

\[\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}\]

Half-angle identities

\[\sin\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1-\cos\theta}{2}}\]

\[\cos\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1+\cos\theta}{2}}\]

\[\tan\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}} = \dfrac{\sin\theta}{1+\cos\theta} = \dfrac{1-\cos\theta}{\sin\theta}\]

Reduction formulas

\[\sin^2\theta=\dfrac{1-\cos2\theta}{2}\]

\[\cos^2\theta=\dfrac{1+\cos2\theta}{2}\]

\[\tan^2\theta=\dfrac{1-\cos2\theta}{1+\cos2\theta} = \dfrac{\sin2\theta}{1+\cos2\theta} = \dfrac{1-\cos2\theta}{\sin2\theta}\]