Double Angle and Half Angle Formulas

Recovering the Double Angle Formulas

Using the sum formula and difference formulas for Sine and Cosine we can observe the following identities:

$\sin(2\theta )=2\sin(\theta )\cos(\theta )$

$\cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )$

Using the Pythagorean Identities for trigonometric functions we can also see that:

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

$\cos(2\theta )=2\cos ^{2}(\theta )-1$

Half Angle Formulas

Using the last two double angle formulas we can now solve for the half angle formulas:

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

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

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