Trigonometric Formulas

Angle Sum and Difference Identities

\sin (A+B)=\sin A \cos B+\cos A \sin B

\sin (A-B)=\sin A \cos B-\cos A \sin B

\cos (A+B)=\cos A \cos B-\sin A \sin B

\cos (A-B)=\cos A \cos B+\sin A \sin B

\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}

\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}

Double Angle Formulas

\sin 2 A=2 \sin A \cos A=\frac{2 \tan A}{1+\tan ^2 A}

\cos 2 A=2 \cos ^2 A-1=1-2 \sin ^2 A=\cos ^2 A-\sin ^2 A=\frac{1-\tan ^2 A}{1+\tan ^2 A}

\tan 2 A=\frac{2 \tan A}{1-\tan ^2 A}

Triple Angle Formulas

\sin 3 A=3 \sin A-4 \sin ^3 A

\cos 3 A=4 \cos ^3 A-3 \cos A

\tan 3 A=\frac{3 \tan A-\tan ^3 A}{1-3 \tan ^2 A}

Sum to Product Identities

\cos C+\cos D=2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)

\cos C-\cos D=-2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)

\sin C+\sin D=2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)

\sin C-\sin D=2 \sin \left(\frac{C-D}{2}\right) \cos \left(\frac{C+D}{2}\right)

Product to Sum Identities

2 \cos A \cos B=\cos (A+B)+\cos (A-B)

-2 \sin A \sin B=\cos (A+B)-\cos (A-B)

2 \sin A \cos B=\sin (A+B)+\sin (A-B)

2 \cos A \sin B=\sin (A+B)-\sin (A-B)