Find the exact value by using a half-angle identity. Sine of seven pi divided by eight.

Solution:

Given, Sine of seven pi divided by eight i.e. \(sin(\frac{7\pi }{8})\)

We have to find the exact value by using a half-angle identity of \(sin(\frac{7\pi }{8})\)

\(sin(\frac{7\pi}{8})=sin(-\frac{\pi }{8}+2\pi )=-sin(\frac{\pi }{8})\)

Now, find \(sin(\frac{\pi }{8})\) by using trigonometric identity,

\(cos(2a)=1-2sin^{2}a\)

\(cos(2(\frac{\pi }{8}))= cos(\frac{\pi }{4})\)

We know, \(cos(\frac{\pi }{4})=\frac{\sqrt{2}}{2}\)

So, \(\frac{\sqrt{2}}{2}=1-2sin^{2}(\frac{\pi }{8})\)

On rearranging,

\(2sin^{2}(\frac{\pi }{8})=1-\frac{\sqrt{2}}{2}\)

\(2sin^{2}(\frac{\pi }{8})=\frac{2-\sqrt{2}}{2}\)

\(sin^{2}(\frac{\pi }{8})=\frac{2-\sqrt{2}}{4}\)

Taking square root,

\(sin(\frac{\pi }{8})=\sqrt{\frac{2-\sqrt{2}}{4}}\)

\(sin(\frac{\pi }{8})=\frac{\sqrt{{2-\sqrt{2}}}}{2}\)

The negative answer is rejected because \(sin(\frac{\pi }{8})\) is positive.

Therefore, the exact value of \(sin(\frac{7\pi }{8})\)=\(\frac{\sqrt{{2-\sqrt{2}}}}{2}\).

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Find the exact value by using a half-angle identity. Sine of seven pi divided by eight.

Summary:

The exact value of sine of seven pi divided by eight by using a half-angle identity is \(\frac{\sqrt{{2-\sqrt{2}}}}{2}\).