## anonymous one year ago What is the exact value of cos 17pi/8? a) square root of 2+the square root of 2/4 b.) 0.38 c.) 0.99 d.)square root of 2-the square root of 2/4 ***My Answer: D***

1. Loser66

Nope

2. anonymous

Dangit hahaha. My second choice would have to be A then

3. Loser66

Nope again, ahhahha

4. anonymous

Dangit(lol)...Not the best at trig identities. Sorry

5. Loser66

ok, hint 17pi/8 = 16pi/8 + pi/8

6. Loser66

and you have the perfect angle 16pi/8 = 2pi

7. Loser66

break it out by cos (a+b) =....

8. anonymous

How would I find the a and b?

9. anonymous

Would it just be a= 16pi/8 and b= pi/8?

10. Loser66

hey, $$cos (\dfrac{17\pi}{8}=cos (\dfrac{16\pi +\pi}{8}$$

11. Loser66

yup

12. anonymous

Oh I see. Ok

13. anonymous

So from there we get it into the form...$\cos(\frac{ 16\pi }{ 8 })\cos(\frac{ \pi }{ 8 }) + \sin(\frac{ 16\pi }{ 8 })\sin(\frac{ \pi }{ 8 })$

14. anonymous

I think

15. anonymous

@Loser66 Now this is the part that I have trouble understanding. I do not know where to go from here

16. Loser66

whyyyyyyyyy? 16pi/8 = 2pi, right?

17. anonymous

Yes

18. Loser66

cos 2pi =1, right?

19. Loser66

sin 2pi =0, ok?

20. anonymous

Ok

21. Loser66

so, at the end, you just have cos (pi/8 ) =0.99999

22. anonymous

Ohhhhhh ok! That was my problem all along! I would divide pi/8 without the cos! Haha Thank you so much!

23. Loser66

ok

24. anonymous

What the... my computer just told me that it was A :(

25. Loser66

Facepalm. hehehe... let's get other's help @campbell_st

26. mathstudent55

$$\dfrac{17\pi}{8} = \dfrac{16\pi}{8} + \dfrac{\pi}{8} = 2\pi + \dfrac{\pi}{8}$$ $$\cos \dfrac{17\pi}{8} = \cos 2\pi + \dfrac{\pi}{8} = \cos \dfrac{\pi}{8}$$ $$\Large \cos \dfrac{\pi}{8} = \cos \left ( {\dfrac{\frac{\pi}{4}}{2} } \right)$$ $$\cos \dfrac{\theta}{2} = \sqrt{\dfrac{1}{2}(1 + \cos \theta) }$$ $$\Large \cos \dfrac{\pi}{8} = \cos \left ( {\dfrac{\frac{\pi}{4}}{2} } \right) = \sqrt{\dfrac{1}{2}(1 + \cos \dfrac{\pi}{4})}$$ $$= \sqrt{\dfrac{1}{2} (1 + \dfrac{\sqrt{2}}{2})} = {\sqrt{\dfrac{1}{2} + \dfrac{\sqrt{2}}{4}}}$$ $$= \sqrt{\dfrac{2 + \sqrt 2}{4}} = \dfrac{\sqrt{2 + \sqrt 2 } }{ 2 }$$

27. Loser66

WWWWWWWWWWWWoooooah!!! It is much.... more logic than my way. Thank you so much @mathstudent55

28. mathstudent55

You're welcome.

29. mathstudent55

@Loser66 You were trying to use the identity for the cosine of a sum of angles. That method does not work in this problem. That method works if you can break up the angle into two angles whose cosines you know. The problem here is that the cos of pi/8 is not one of the know values. On the other hand, the cos of pi/4 is well known, so using the identity of the cosine of a half angle gives you the result.