Trigonometry

Trig Identities

Pythagorean Identities

\[\sin^2(x) + \cos^2(x) = 1\] \[\csc^2(x) - \cot^2(x) = 1\] \[\sec^2(x) - \tan^2(x) = 1\]

Double Angle Identities

\[\sin(2x) = 2\sin(x)\cos(x)\] \[\cos(2x) = \cos^2(x) - \sin^2(x) = 1-2\sin^2(x) = 2\cos^2(x)-1\] \[\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\]

Half Angle Identities

\[\sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos(x)}{2}}\] \[\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos(x)}{2}}\] \[\tan\left(\frac{x}{2}\right) = \frac{1-\cos(x)}{\sin(x)} = \frac{\sin(x)}{1+\cos(x)}\]

Angle Sum Identities

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

Angle Different Identities

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

Products as Sums

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

Sums as Products

\[\sin(x) + \sin(y) = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\] \[\sin(x) - \sin(y) = 2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\] \[\cos(x) + \cos(y) = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\] \[\cos(x) - \cos(y) = -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\]

Power Reducing Formulas

\[\sin^2(x) = \frac{1-\cos(2x)}{2}\] \[\cos^2(x) = \frac{1+\cos(2x)}{2}\] \[\tan^2(x) = \frac{1-\cos(2)}{1+\cos(2)}\]