Trigonometric Formulae

Half Angle Formulas

$\sin\frac{A}{2}=\pm\sqrt{\frac{1-\cos A}{2}}$

+ if $\frac{A}{2}$ lies in quadrant 1 or 2

- if $\frac{A}{2}$ lies in quadrant 3 or 4

$\cos\frac{A}{2}=\pm\sqrt{\frac{1+\cos A}{2}}$

+ if $\frac{A}{2}$ lies in quadrant 1 or 4

- if $\frac{A}{2}$ lies in quadrant 2 or 3

$\cot\frac{A}{2}=\pm\sqrt{\frac{1+\cos A}{1-\cos A}}$

+ if $\frac{A}{2}$ lies in quadrant 1 or 3

- if $\frac{A}{2}$ lies in quadrant 2 or 4

$\tan\frac{A}{2}=\pm\sqrt{\frac{1-\cos A}{1+\cos A}}$

+ if $\frac{A}{2}$ lies in quadrant 1 or 3

- if $\frac{A}{2}$ lies in quadrant 2 or 4

$\cot\frac{A}{2} = \frac{\sin A}{1-\cos A} = \frac{1+\cos A}{\sin A}=\csc A+\cot A$

$\tan\frac{A}{2} = \frac{\sin A}{1+\cos A} = \frac{1-\cos A}{\sin A}=\csc A-\cot A$

Power Reducing Formulae

$\sin^2(A)=\frac{1 - \cos(2A)}{2}$

$\sin^3(A)=\frac{3\sin A - \sin(3A)}{4}$

$\sin^4(A)=\frac{\cos(4A) - 4\cos(2A) + 3}{8}$

$\cos^2(A) = \frac{1 + \cos(2A)}{2}$

$\cos^3(A)=\frac{3\cos A + \cos(3A)}{4}$

$\cos^4(A)=\frac{4\cos(2A) + \cos(4A) + 3}{8}$

Tangent half-angle substitution

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

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

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

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

Sum and Difference of Angles

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

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

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

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

$\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)}=\frac{\sin(A)\cdot \cos(B) + \cos(A)\cdot \sin(B)}{\cos(A)\cdot \cos(B) - \sin(A)\cdot \sin(B)}$

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

$\cot(A \pm B) = \frac{\cot(B)\cot(A)\mp 1}{\cot(B)\pm \cot(A)}=\frac{1\mp \tan(A)\tan(B)}{\tan(A)\pm \tan(B)}$

$\sin(A + B + C) = \sin A\cdot\cos B\cdot\cos C + \cos A\cdot\sin B\cdot\cos C + \cos A\cdot\cos B\cdot\sin C - \sin A\cdot\sin B\cdot\sin C$

$\cos(A + B + C) = \cos A\cdot\cos B\cdot\cos C - \sin A\cdot\sin B\cdot\cos C - \sin A\cdot\cos B\cdot\sin C - \sin A\cdot\cos B \cdot\sin C - \cos A \cdot \sin B\cdot \sin C$

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

Double And Triple Angle Formulae

$\sin(2A) = 2\sin(A)\cdot \cos(A)$

$\cos(2A) = \cos^2(A) - \sin^2(A) = 2\cos^2(A) - 1 = 1 - 2\sin^2(A)$

$\tan(2A) = \frac{2\tan(A)}{1- \tan^2(A)}$

$\cos(2A) = \frac{1 - \tan^2(A)}{1 + \tan^2(A)}$

$\sin(2A) = \frac{2\tan(A)}{1 + \tan^2(A)}$

$\sin3A = 3\sin A - 4 \sin^3A$

$\cos3A = 4\cos^3A - 3 \cos A$

$\tan3A=\frac{3\tan A - \tan^3A}{1-3\tan^2A}$

$\cot3A=\frac{\cot^3A-3\cot A}{3\cot^2A-1}$

$\sin4A = 4\cos^3A\cdot \sin A - 4\cos A\cdot \sin^3A$

$\cos4A = \cos^4A - 6\cos^2A\cdot \sin^2A + \sin^4A$

$\tan4A=\frac{4\tan A - 4\tan^3A}{1-6\tan^2A+\tan^4A}$

$\cot4A=\frac{\cot^4A-6\cot^2A+1}{4\cot^3A-4\cot A}$

Sum and Difference of Trigonometric Functions

$\textrm{ sin }A \textrm{ sin }B = \frac{1}{2} (\textrm{ cos }(A - B) - \textrm{ cos }(A + B))$

$\textrm{ cos }A \textrm{ cos }B = \frac{1}{2} (\textrm{ cos }(A - B) + \textrm{ cos }(A + B))$

$\textrm{ sin }A \textrm{ cos }B = \frac{1}{2} (\textrm{ sin }(A + B) + \textrm{ sin }(A - B))$

$\tan A \cdot \tan B = \frac{\tan A+\tan B}{\cot A+\cot B}=-\frac{\tan A-\tan B}{\cot A-\cot B}$

$\cot A \cdot \cot B = \frac{\cot A+\cot B}{\tan A+\tan B}$

$\tan A \cdot \cot B = \frac{\tan A+\cot B}{\cot A+\tan B}$

$\sin A\sin B\sin C = \frac{1}{4}\big(\sin(A+B-C)+\sin(B+C-A)+\sin(C+A-B)-\sin(A+B+C)\big)$

$\cos A\cos B\cos C = \frac{1}{4}\big(\cos(A+B-C)+\cos(B+C-A)+\cos(C+A-B)+\cos(A+B+C)\big)$

$\sin A\sin B\cos C = \frac{1}{4}\big(-\cos(A+B-C)+\cos(B+C-A)+\cos(C+A-B)-\cos(A+B+C)\big)$

$\sin A\cos B\cos C = \frac{1}{4}\big(\sin(A+B-C)-\sin(B+C-A)+\sin(C+A-B)+\sin(A+B+C)\big)$

Multiplication of 2 Trigonometric Functions

$\textrm{ sin }A \textrm{ sin }B = \frac{1}{2} (\textrm{ cos }(A - B) - \textrm{ cos }(A + B))$

$\textrm{ cos }A \textrm{ cos }B = \frac{1}{2} (\textrm{ cos }(A - B) + \textrm{ cos }(A + B))$

$\textrm{ sin }A \textrm{ cos }B = \frac{1}{2} (\textrm{ sin }(A + B) + \textrm{ sin }(A - B))$

$\tan A \cdot \tan B = \frac{\tan A+\tan B}{\cot A+\cot B}=-\frac{\tan A-\tan B}{\cot A-\cot B}$

$\cot A \cdot \cot B = \frac{\cot A+\cot B}{\tan A+\tan B}$

$\tan A \cdot \cot B = \frac{\tan A+\cot B}{\cot A+\tan B}$

$\sin A\sin B\sin C = \frac{1}{4}\big(\sin(A+B-C)+\sin(B+C-A)+\sin(C+A-B)-\sin(A+B+C)\big)$

$\cos A\cos B\cos C = \frac{1}{4}\big(\cos(A+B-C)+\cos(B+C-A)+\cos(C+A-B)+\cos(A+B+C)\big)$

$\sin A\sin B\cos C = \frac{1}{4}\big(-\cos(A+B-C)+\cos(B+C-A)+\cos(C+A-B)-\cos(A+B+C)\big)$

$\sin A\cos B\cos C = \frac{1}{4}\big(\sin(A+B-C)-\sin(B+C-A)+\sin(C+A-B)+\sin(A+B+C)\big)$

Other Trigonometric Functions

$1\pm\sin A=2\sin^2\big(\frac{\pi}{4}\pm \frac{A}{2}\big)=2\cos^2\big(\frac{\pi}{4}\mp \frac{A}{2}\big)$

$\frac{1-\sin A}{1+\sin A} = \tan^2(\frac{\pi}{4}-\frac{A}{2})$

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

$\frac{1-\tan A}{1+\tan A} = \tan(\frac{\pi}{4}-A)$

$\frac{1+\tan A}{1-\tan A} = \tan(\frac{\pi}{4}+A)$

$\frac{\cot A + 1}{\cot A - 1} = \cot(\frac{\pi}{4}-A)$

$\tan A + \cot A = \frac{2}{\sin2A}$

$\tan A - \cot A = -2\cot2A$