Question

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

sin ( 7 pi / 8 )

Answer 1

Do you know what the half angle identity is?

Answer 2

Yeah

Answer 3

sin(7pi/8) = sin((7pi/4)/2), right?

sin(u/2) = sqrt((1-cos(u)/2) = sqrt((1-cos(7pi/4))/2

8pi/4 is a full circle. 7pi/4 is 45° below the x axis, so cos(7pi/4) = sqrt(2/2), so

sin(7pi/8) = sqrt((1-sqrt(2)/2)/2) = sqrt(1/2-sqrt(2)/4) = 0.38

Since 7pi/8 is above the x axis, the sin is positive, so the full answer is

sin(7pi/8) = +0.38. Plugging into my calculator and computing the sin directly, I confirm that this answer is correct.

Answer 4

\[\cos 2x = 1-2 \sin^{2} x\]
\[\cos \frac{7\pi}{4} = 1-2\sin^{2} \frac{7\pi}{8}\]
\[\frac{\sqrt{2}}{2} = 1-2\sin^{2} \frac{7\pi}{8}\]
\[2\sin^{2} \frac{7\pi}{8} = 1-\frac{\sqrt{2}}{2} = \frac{2-\sqrt{2}}{2}\]
\[\sin \frac{7\pi}{8} = \frac{\sqrt{2-\sqrt{2}}}{2}\]

Answer 5

Why do people ask such dumb questions, this doesn't help anything