Question

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

sine of seven pi divided by eight.

Answer 1

\[\LARGE \sin \left(\frac{7\pi}{8}\right)=\sin\left(\frac{\color{red}{\frac{7\pi}{4}}}{2}\right)\]

Answer 2

That give you any hints? :)

Answer 3

uhnm im just not sure how to solve this one

Answer 4

Well, I have to go now, here's all you need to know:
\[\Large \sin\left(\frac{7\pi}{4}\right)=-\frac{\sqrt2}{2}\]
\[\Large \sin\left(\frac{\theta}2\right)= \pm\sqrt{\frac{1-\cos(\theta)}{2}}\]
plus or minus depending on which quadrant \(\Large \frac{7\pi}8\) lies...

Answer 5

im just confused on what to plug in to figure the problem out and get a answer

Answer 6

Whoops....
\[\Large \cos\left(\frac{7\pi}{4}\right)= \frac{\sqrt2}{2}\]

Answer 7

All you need to do is plug in :D

Answer 8

is its square root of 2 over 2 and square root of -2 over two because that's what 7pieover 4 is equal to on unit circle?

Answer 9

lol im so sorry im just so lost on this dumb question

Answer 10

\[\Large \sin\left(\frac{\theta}2\right)= \pm\sqrt{\frac{1-\cos(\theta)}{2}}\]

Answer 11

\[\LARGE \sin \left(\frac{7\pi}{8}\right)=\sin\left(\frac{\color{red}{\frac{7\pi}{4}}}{2}\right)\]\[\LARGE \sin\left(\frac{\color{red}{\frac{7\pi}{4}}}{2}\right)= \pm\sqrt{\frac{1-\cos(\color{red}{\frac{7\pi}{4}})}{2}}\]

Answer 12

what does that mean though like am I factoring or what like I see all these eqautions and stuff I see where seven pi divided by 8 comes from the original problem I just dont know what to plug in

Answer 13

It means... I just gave you the exact value of \(\Large \cos \frac{7\pi}{4}\)

Answer 14

So you can just plug it in here:
\[\LARGE \sin\left(\frac{\color{red}{\frac{7\pi}{4}}}{2}\right)= \pm\sqrt{\frac{1-\boxed{\cos(\color{red}{\frac{7\pi}{4}})}}{2}}\]

Answer 15

ok so then I plug that into calculator?

Answer 16

Calculator? No.
Just replace the thing I boxed with its value
\[\Large \frac{\sqrt2}{2}\] and solve.

Answer 17

ok thank you