1. finding the exact value..

well this is one problem i have no idea how to start. i tried using the sum of angles but i cannot find anything that would help me get a start on the problem. im asked to find the exact value of cos 36? i thought about double angle identities..i cannot think of anything that would work for this problem. any help would be appreciated. thx in advance...

2. Originally Posted by slapmaxwell1
well this is one problem i have no idea how to start. i tried using the sum of angles but i cannot find anything that would help me get a start on the problem. im asked to find the exact value of cos 36? i thought about double angle identities..i cannot think of anything that would work for this problem. any help would be appreciated. thx in advance...
Good start with the double angle identities, but you're not thinking big enough:

Try solving $\displaystyle \cos 5\theta = -1$

3. well if i was solving for theta, cos 5theta = -1 then i would say con inverse of (-1) = 5 theta, and then divide both sides by 5 to get theta, but i dont see the connection? im still missing the big picture....oh and yeah i tried the double angle, i used double angle to solve and figure out with cos 15 was...the 36 is really throwing me off, maybe if i add it something and then subtract it, i could try to use a difference formula...one of the guys in class said something about using the quadratic equation to solve the problem??? i really dont see the connection there...

4. Originally Posted by Gusbob
Good start with the double angle identities, but you're not thinking big enough:

Try solving $\displaystyle \cos 5\theta = -1$
ok wait now that i thought about it a little more, i would break it apart and make it the sum of angles..cos (4theta + theta) = -1 then i use the add properties and break it apart and so on...once i broke it apart i would have to use some substitution for the 4 theta..but i still dont see the connection...

5. Originally Posted by slapmaxwell1
well if i was solving for theta, cos 5theta = -1 then i would say con inverse of (-1) = 5 theta, and then divide both sides by 5 to get theta, but i dont see the connection? im still missing the big picture....oh and yeah i tried the double angle, i used double angle to solve and figure out with cos 15 was...the 36 is really throwing me off, maybe if i add it something and then subtract it, i could try to use a difference formula...one of the guys in class said something about using the quadratic equation to solve the problem??? i really dont see the connection there...
Sorry, I must have been too vague. In fact this question is more difficult than I first suspected.

I was trying to get you to realise that 36 x 5 = 180.

Let $\displaystyle \theta = 36$ degrees

So we have $\displaystyle \cos 5 \theta = \cos (5 \times 36) = \cos 180 = -1 \Rightarrow \cos 5\theta + 1 = 0$

We also (well I also) know that $\displaystyle \cos 5 \theta = 16 \cos^5 \theta -20 \cos ^3 \theta+5\cos \theta$ (you get this by angle sums like you've tried. I'll recommend cos (3x+2x), more efficient).

So letting x = cos $\displaystyle \theta$ = cos 36 we have

$\displaystyle 16x^5 - 20x^3 + 5x + 1 = 0$ which factorises to

$\displaystyle (x+1)(4x^2-2x-1)^2 = 0$

Can you continue?