Finding cos(t) from cos(2t) = 1/3 using double angle identities

[hotrocks007]

I'm taking Calculus next year and over the summer I have some assignments.
This one is due in a couple of hours, so any help would be appreciated!

If cos2t=1/3 and *0<_ 2t <_ pie, find cost. t=theta *less than or equal to

I don't know how to use the identities to help me.

Please help!

 

[Jameson]

Well...

$$\cos{(2\alpha)} = 2\cos^2{(\alpha)}-1$$

Plug that in and post when you make progress.

 

[hotrocks007]

ohh, costheta=sqrt(6)/3
?

 

[Jameson]

$$2\cos^2{(\alpha)}-1 = \frac{1}{3}$$

$$2\cos^2{(\alpha)}=\frac{4}{3}$$

$$\cos^2{(\alpha)}=\frac{2}{3}$$

Can you finish from here?

 

[hotrocks007]

oh yes thanks!

how about this one.

I'm not sure how to simplify it down, and how to distribute the ^2 once it has been plugged in.
x^2 + y^2 +3x=0 when x=rcostheta and y=rsintheta

 

[Nylex]

Remember that $\sin^2 x + \cos^2 x = 1$. These questions don't seem to have anything to do with calculus, they just seem to be trigonometry.

 

[Jameson]

$$(r\cos{\theta})^2 + (r\sin{\theta})^2+3(r\cos{\theta})=0$$

$$r^2\cos^2{(\theta)}+r^2\sin^2{(\theta)}+3(r\cos{\theta})=0$$

Do you see the trig identity coming in?

 

[hotrocks007]

the Pythag. Identity? Would you have to plug in rcostheta with the 3x?

 

[Jameson]

I should have plugged that in earlier. But no, that's not where the identity comes in.

I'll give you my last hint to this problem.

$$r^2\cos^2{(\theta)}+r^2\sin^2{(\theta)}+3(r\cos{\theta})=0$$

$$r^2(\cos^2{\theta}+\sin^2{\theta})...$$

 

[hotrocks007]

OH! thanks!

 

[hotrocks007]

when you distribute the 3, would it be 3rcos3theta? or do you just not distribute the 3 to the cos?

 

[Jameson]

$$3r\cos{(\theta)}$$