Question

Find the exact value by using a half-angle identity.

cosine of five pi divided by twelve.

Answer 1

Answer on Question #48679 – Math – Trigonometry

Find the exact value by using a half-angle identity.

cosine of five pi divided by twelve.

Solution:

We need to find the exact value of the

$$\cos \frac {5 \pi}{12}$$

Using a half-angle identity:

$$\cos \frac {5 \pi}{12} = \cos \frac {\frac {5 \pi}{6}}{2} = \pm \sqrt {\frac {1}{2} \left(1 + \cos \frac {5 \pi}{6}\right)}$$

We know that $\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$

$$\begin{array}{l} \cos \frac {5 \pi}{12} = \pm \sqrt {\frac {1}{2} \left(\frac {2}{2} - \frac {\sqrt {3}}{2}\right)} \\ \cos \frac {5 \pi}{12} = \pm \sqrt {\frac {1}{2} \left(\frac {2 - \sqrt {3}}{2}\right)} \\ \cos \frac {5 \pi}{12} = \pm \sqrt {\frac {2 - \sqrt {3}}{4}} \\ \cos \frac {5 \pi}{12} = \pm \frac {\sqrt {2 - \sqrt {3}}}{2} \\ \end{array}$$

$\frac{5\pi}{12}$ is in quadrant 1 so we choose the «+» sign because cosine is positive there

$$\cos \frac {5 \pi}{12} = \frac {\sqrt {2 - \sqrt {3}}}{2}$$

Answer: $\cos \frac{5\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2}$

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