How to use the half angle identity to find exact value of Sin (3pi/8)?

Question

Brady Mays · Accepted Answer

This approach to solving this kind of math problem is common.
Call

\sin a = \sin (\frac{3 π}{8})

\cos 2 a = \cos (\frac{6 π}{8}) = \cos (\frac{3 π}{4}) = - \frac{\sqrt{2}}{2}

Apply the trig identity:

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

- \frac{\sqrt{2}}{2} = 1 - 2 \sin^{2} a

2 \sin^{2} a = 1 + \frac{\sqrt{2}}{2} = \frac{2 + \sqrt{2}}{2}

\sin^{2} a = \frac{2 + \sqrt{2}}{4}

\sin a = \sin (\frac{3 π}{8}) = \pm \frac{\sqrt{2 + \sqrt{2}}}{2} .

Since the arc

(\frac{3 π}{8})

is located in Quadrant I, its sin is positive, then

\sin (\frac{3 π}{8}) = \frac{\sqrt{2 + \sqrt{2}}}{2}

Check by calculator.

\sin (\frac{3 π}{8}) = \sin 67.5 d e g = 0.92

\frac{\sqrt{2 + \sqrt{2}}}{2} = \frac{1.84}{2} = 0.92

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