## anonymous 3 years ago Find sin x/2, cos x/2, and tan x/2, if cos x = -12/13, 180 degrees is less than x which is less than 270 degrees

1. myininaya

|dw:1352334915182:dw| But anyways, mainly I drew this to show what quadrant we are in. sine and cosine are both negative in this quadrant Do you know the half angle identities for sine and cosine?

2. anonymous

I know them but I don't know how to use them.

3. myininaya

$\sin(\frac{x}{2})=\pm \sqrt{\frac{1}{2}(1-\cos(x))}$ $\cos(\frac{x}{2})=\pm \sqrt{\frac{1}{2}(1+\cos(x))}$ $\tan(\frac{x}{2})=\frac{\sin(\frac{x}{2})}{\cos(\frac{x}{2})}$

4. anonymous

I know how to get the decimal answers, but I need exact answers.

5. myininaya

Since we are looking at the third quadrant and I said sine and cosine are both negative there, then we are actually looking at: $\sin(\frac{x}{2})=-\sqrt{\frac{1}{2}(1-\cos(x))}$ $\cos(\frac{x}{2})=-\sqrt{\frac{1}{2}(1+\cos(x))}$ Replace cos(x) with what it equals which is -12/13 then simplify

6. anonymous

For sin x/2 I got -sqrt of 25/26, which is wrong apparently, but I got cos x/2=-sqrt of 1/26 and it was right. Also, how do I find tan x/2 from all of this?

7. myininaya

$\sin(\frac{x}{2})=-\sqrt{\frac{1}{2}(1-\frac{-12}{13})} =- \sqrt{\frac{1}{2}(1+\frac{12}{13})}=-\sqrt{\frac{1}{2}(\frac{13}{13}+\frac{12}{13})}$ $=-\sqrt{\frac{1}{2}(\frac{25}{13})}=-\sqrt{\frac{25}{26}}=-\frac{\sqrt{25}}{\sqrt{26}}=-\frac{5}{\sqrt{26}}$ This answer can be simplified more by choosing to rationalize the denominator

8. anonymous

Ok thanks. I figured out the tan x/2 part

9. myininaya

np.