Trigonometric Identities

Half Angle Identities

$$ \cos{\frac{\theta}{2}} = \sqrt{\frac{1 + \cos{\theta}}{2}} $$

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

therefore:

$$ 2\cos^2{\frac{\theta}{2}} = 1 + \cos{\theta} $$

$$ 2\sin^2{\frac{\theta}{2}} = 1 - \cos{\theta} $$

Sum and Difference Formulas

$$ \cos{(a + b)} = \cos{a}\cos{b}-\sin{a}\sin{b} $$

$$ \cos{(a - b)} = \cos{a}\cos{b}+\sin{a}\sin{b} $$

$$ \sin{(a + b)} = \sin{a}\cos{b}+\cos{a}\sin{b} $$

$$ \sin{(a - b)} = \sin{a}\cos{b}-\cos{a}\sin{b} $$

Linear Combinations

$$ a \cos x + b \sin x = \sqrt{a^2 + b^2} \cos{(x - \arctan2{(b, a)})} $$

$$ a \cos x + b \sin x = \sqrt{a^2 + b^2} \cos{(x + \text{Arctan}{(-b/a))})} $$