Find the exact value by using a half-angle identity. Cosine of five pi divided by twelve.

Solution:

Given, \(cos(\frac{5\pi }{12})\)

We have to find the exact value of cosine\(\frac{5\pi }{12}\) by using a half-angle identity.

By the half angle formula,

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

Here, \(\frac{\theta }{2}=\frac{5\pi }{12}\)

Then, \(\theta =\frac{5\pi }{6}\)

We know that \(\frac{5\pi }{6}\) is a standard angle in quadrant 2 with a reference angle of \(\frac{\pi }{6}\)

So, \(cos(\frac{5\pi }{6})=-cos(\frac{\pi }{6})=-\frac{\sqrt{3}}{2}\)

Therefore,

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

\(cos(\frac{5\pi }{12})=\pm \sqrt{\frac{1-\frac{\sqrt{3}}{2} }{2}}\)

= \(\pm \sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}}\)

= \(\pm \sqrt{\frac{2-\sqrt{3}}{4}}\)

= \(\pm \frac{\sqrt{2-\sqrt{3}}}{2}\)

Since \(\frac{5\pi }{12}< \frac{\pi }{2}\), \(\frac{5\pi }{12}\) is in first quadrant.

So, \(cos(\frac{5\pi }{12})\) is positive

\(cos(\frac{5\pi }{12})\approx 0.26\)

Therefore, the exact value is \(cos(\frac{5\pi }{12})=\frac{\sqrt{2-\sqrt{3}}}{2}\approx 0.26\).

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Find the exact value by using a half-angle identity. Cosine of five pi divided by twelve.

Summary:

The exact value of cosine of five pi divided by twelve by using a half-angle identity is \(cos(\frac{5\pi }{12})=\frac{\sqrt{2-\sqrt{3}}}{2}\approx 0.26\).