20.4 Half-Angle Formulas

If we let $θ = α / 2$ in the identity $\cos 2 θ = 1 - 2 \sin^{2} θ$ and then solve for $\sin (α / 2),$

\sin \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{2}}

(20.26)

Also, with the same substitution in the identity $\cos 2 θ = 2 \cos^{2} θ - 1,$ which is then solved for $\cos (α / 2),$ we have

\cos \frac{α}{2} = \pm \sqrt{\frac{1 + \cos α}{2}}

(20.27)

CAUTION

In each of Eqs. (20.26) and (20.27), the sign chosen depends on the quadrant in which $\frac{α}{2}$ lies.

We can find $\cos 165 °$ by using the relation

\begin{array}{rcll} \cos 165 ° & = & - \sqrt{\frac{1 + \cos 330 °}{2}} & using Eq . (20.27) \\ = & - \sqrt{\frac{1 + 0.8660}{2}} & = - 0.9659 \end{array}

Here, the minus sign is used, since $165 °$ is in the second quadrant, and the cosine of a second-quadrant angle is negative.

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