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:
\[2\cos^2{\frac{\theta}{2}} = 1 + \cos{\theta}\]
\[2\sin^2{\frac{\theta}{2}} = 1 - \cos{\theta}\]
\[\cos{(a + b)} = \cos{a}\cos{b}-\sin{a}\sin{b}\]
\[\cos{(a - b)} = \cos{a}\cos{b}+\sin{a}\sin{b}\]
\[\sin{(a + b)} = \sin{a}\cos{b}+\cos{a}\sin{b}\]
\[\sin{(a - b)} = \sin{a}\cos{b}-\cos{a}\sin{b}\]
Linear Combinations
\[a \cos x + b \sin x = \sqrt{a^2 + b^2} \cos{(x - \arctan2{(b, a)})}\]
\[a \cos x + b \sin x = \sqrt{a^2 + b^2} \cos{(x + \text{Arctan}{(-b/a))})}\]