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Mathematics LibreTexts

Fundamental Trigonometric Identities

"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)

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}