Question

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

tan 7pi/8

Answer 1

So we'll start from here.
\[\huge \tan \left(\frac{7\pi}{8}\right)=\tan \left(\frac{\left(\frac{7\pi}{4}\right)}{2}\right)\]

Answer 2

\[\huge \tan \left(\frac{a}{2}\right)=\frac{1-\cos a}{\sin a}\]Using this identity gives us,

Answer 3

\[\huge \frac{1-\cos \left(\frac{7\pi}{4}\right)}{\sin \left(\frac{7\pi}{4}\right)}\]And from here, you just need to remember your special angles on the unit circle.

Answer 4

and that is?

Answer 5

cos(7pi/4) and the sine also, you need to remember those. If you don't know, go back to your chart or whatever you use to figure out the special angles :D

It's one of the pi/4 angles, so it'll be something involving sqrt(2)/2's

Answer 6

https://www.youtube.com/watch?v=Nh4Ro86dnHk