Question

find the following.

Answer 1

given that \[ \pi \le \theta \le \pi\] and that \[\cos \theta = -12/13, \] find

\[\sin \theta \]

Answer 2

is it 5/13?

Answer 3

5/13 seems correct to me.

Answer 4

It's either 5/13 or -5/13, and if it were negative, then \(\theta\) wouldn't be in our required range. Thus, \(\theta =5/13\)

Answer 5

Actually, could you be more specific about your range? Is it \[0 \leq \theta \leq \pi\]Or \[-\pi\leq\theta\leq\pi\]

Answer 6

how do i find \[\cos 2\theta\]

Answer 7

actually its \[\pi/2 \le \theta \le \pi \]

Answer 8

Then our answer is 5/13.

As for \(\cos(2\theta)\), use the identity\[\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)\]

Answer 9

If you want verification for an answer, I'm getting \[119 \over 169\]

Answer 10

could i also say that \[\cos 2\theta = 1- 2 \sin ^{2} \theta \]

Answer 11

In which case we would get \[-119 \over 169\]However, now we have a different domain, \[\pi\leq 2\theta\leq2\pi\]

Answer 12

how did you get 119/169 ?

Answer 13

\[{144 \over 169}-{25 \over 169}\]

Answer 14

\[\cos \theta = -12/13 \] and \[\sin \theta = 5/13\]

do you just plug those in to the identty?

Answer 15

That's exactly what I did. Also,fIf we imagine the unit circle, I'm pretty sure \[119 \over 169\]is the correct answer.

Answer 16

is \[\sin(\theta + \pi ) \] = -5/13 ?

Answer 17

I believe so.

Answer 18

Thank you!

Answer 19

You're very welcome.