Question

Find the function value.

cos^2(7pi/8)=

Answer 1

Some sort of half-angle formula applies here.

Answer 2

Ok what should I do?

Answer 3

Mhm, absolutely. Notice that we do not have a common unit circle value for any division of 8. But we do have a value for is 7pi/4. Now in order to actually get the value of 7pi/8, we use a half angle formula. So what we do is we will write out the formula and plug 7pi/4 into that formula, this will give us the value of 7pi/8. So the formula we need to plug 7pi/4 into is this:

\[\pm \sqrt{\frac{ 1+\cos2\theta }{ 2 }} \]SO if you plug in 7pi/4 for theta in that formula, you think you could simplify it to find 7pi/8?

Answer 4

I apologize, looks like only theta, not 2 theta.

Answer 5

\[
\cos^2\left(\frac \theta 2\right) = \frac{1+\cos(\theta)}2
\]

Answer 6

Yeah, once you square it you would have that, lol. I feel bad, still need to correct my mistake in what I posted

\[\pm \sqrt{\frac{ 1+\cos \theta }{ 2 }} \]And then yes, wio shortened it by squaring it.

Answer 7

\[
\frac{7\pi}8 = \frac 1 2\left(\frac{7\pi}4\right)
\]In this case I would say \(\theta = 7\pi/4\)

Answer 8

so the final answer is 7pi/4? My Math Lab website says it is wrong

Answer 9

No, I'm saying \[
\cos^2\left(\frac{7\pi}8\right)= \frac{1-\cos(7\pi/4)}2
\]Now you have to find \(\cos(7\pi/4)\).

Answer 10

Now we would use: \[
\cos(\theta - 2\pi) = \cos(\theta)
\]

Answer 11

so the answer is 1/(square root 2) ?

Answer 12

How did you get that?

Answer 13

cos(7pi/4) = 1/(square root 2)

Answer 14

Okay well if you say that, then we have \[
\frac{1-\frac {1} {\sqrt{2}}}{2}
\]And you have to simplify it.

Answer 15

ok

Answer 16

would you show me how to simplify that

Answer 17

It becomes \[
\frac 1 2 - \frac 1 {2\sqrt{2}}
\]You want to simplify the radical: \[
\frac 1 2 - \frac {\sqrt{2}} {2}

\]Then it is simplified as: \[
\frac {1-\sqrt{2}} 2
\]

Answer 18

hmm for some reason that's wrong idk why

Answer 19

Okay how about \[
\frac {1+\sqrt{2}} 2
\]

Answer 20

are you sure? I only have one attempt left :/

Answer 21

http://www.wolframalpha.com/input/?i=cos^2%287pi%2F8%29

Answer 22

@wio

Answer 23

@cinar how did you use wolfarm alpha to get the answer? it does not show anything related to this

Answer 24

can you send screen shot..

Answer 25

here you go

Answer 26

wolfram says cos^2(7pi/8)=0.8535533905932737622004221810524245196424179688442370...

or
cos^2(7pi/8)= \[
\frac{1}{4} (2+\sqrt{2})
\]exactly

Answer 27

got it..

Answer 28

cos(2x)=2cos^2x-1

cos^2x=(cos2x+1)/2
x=7pi/8
cos^2(7pi/8)=(cos7pi/4+1)/2

need to find cos(7pi/4)

pi=180 degree

cos(7pi/4)=cos(7*180/4)=cos(315)

cos(315)=cos(360-45)=cos360cos45+sin360sin45=1*1/sqrt2+0*1/sqrt2=1/sqrt2

cos^2(7pi/8)=(cos7pi/4+1)/2=(1/sqrt2+1)/2=\[ \frac{1}{4} (2+\sqrt{2})\]

Answer 29

thank you @cinar yours is right

Answer 30

yw..

Answer 31

@wio thank you too ^^ you made an effort