Answer:
Step-by-step explanation:
You have to use the half angle identities for trig. You will also need access to a unit circle with radian measures of angles. The identity for the half angle for sin is
[tex]sin(\frac{\theta}{2})=[/tex]±[tex]\sqrt{\frac{1-cos\theta}{2} }[/tex]
To use a half angle identity, the trick is to rewrite the half angle so that when you bring up that 2 in the denominator and flip it to multiply you get back the angle you started with. Our angle is 7π/8. Divide the 8 by 2 and you get your angle to be in the form [tex]sin(\frac{\frac{7\pi}{4} }{2})[/tex]
Note that when you bring up the 2 and flip it to multiply you get back 7π/8.
Do you see that? For the formula, then, our angle theta is 7π/4. If you look to the unit circle, you see that the cosine of that angle is √2/2. Filling in the formula:
[tex]sin(\frac{\frac{7\pi}{4} }{2})=+/-\sqrt{\frac{1-\frac{\sqrt{2} }{2} }{2} }[/tex]
Get a common denominator up on top there and simplify:
[tex]+/-\sqrt{\frac{\frac{2-\sqrt{2} }{2} }{2}[/tex]
You'll bring up that bottom 2 and flip it to multiply:
[tex]+/-\sqrt{\frac{2-\sqrt{2} }{2}*\frac{1}{2} }[/tex]
Simplify to get
[tex]+/-\sqrt{\frac{2-\sqrt{2} }{4} }[/tex]
Since 4 is a perfect square, the final simplification will be to pull it out as a 2:
[tex]sin(\frac{7\pi}{8})=[/tex]±[tex]\frac{\sqrt{2-\sqrt{2} } }{2}[/tex]
And you're all done! Easy if you know how to use the formula and the unit circle.
