Half Angles (C)

It was shown in the last section that

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

and

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

If we let $A = \theta/2$ and solve for sine or cosine of $\theta$ we get the half angle identities:

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

and

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

If the quadrant of $\theta/2$ is known, and the cosine of $\theta$ is known, then the sine and cosine of $\theta/2$ can be computed.

Exercises:

1.

Derive the half angle formulae.

2.

Find the sine and cosine of $\pi/8$.

3.

Find the sine and cosine of $\pi/12$.