Find the exact value by using a half-angle identity. sin 5π/12.

Solution:

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

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

By the half angle formula,

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

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,\(sin(\frac{5\pi }{12}) =sin(\frac{1}{2}(\frac{5\pi }{6}))\\=\sqrt{\frac{1-cos(\frac{5\pi }{6})}{2}}\)

We know, cos(5π/6) = √3/2

So, \(sin(\frac{5\pi }{12})=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}\)

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

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

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

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

Therefore, \(sin(\frac{5\pi }{12})=\frac{\sqrt{2-\sqrt{3}}}{2}\).

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Find the exact value by using a half-angle identity. sin 5π/12.

Summary:

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