How do you solve
\[\cos 2 x= cos x +2\]

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- anonymous

How do you solve
\[\cos 2 x= cos x +2\]

- katieb

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- phi

do you know cos(x+y) formula?

- anonymous

Yes cosxcosy+sinxsiny

- phi

almost, the add turns into a minus (it's ugly that way)
if you use that with y= x
cos(x+x)= cos^2 x - sin^2 x
replace sin^2 with 1-cos^2 : cos^2 - 1 - cos^2 or
2cos^2(x) -1

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## More answers

- phi

you can replace the cos(2x) with 2 cos^2(x) -1
if you call cos(x) y you have
2y^2 -1 = y+2
solve for y

- phi

does that make sense?

- anonymous

I think. I will spend some time with it now. Thanks.

- phi

of course, after you find y, you find x using
x = inverse cos(y)
because this is a quadratic, you will get two answers for y
then we have to figure out the the answers for every 2pi

- phi

*typo
replace sin^2 with 1-cos^2 : cos^2 - (1 - cos^2)= cos^2 -1 + cos^2 or
2cos^2(x) -1

- anonymous

I think I have it! Hang on while I try to finish!!

- anonymous

I got y=3/2 and y=-1. So -1 is at pi but what about 3/2. My teacher says pi is the only answer? Why?

- phi

the cosine goes between -1 and +1 (it never goes outside that range) . There is no angle whose cos is > 1 so you can't get 3/2

- anonymous

Of course! Thanks for helping me see.

- phi

and because the cosine repeats, you can add any multiple of 2pi to your answer and it will work. to be complete the answer is
\[ \pi + 2\pi n \]where n is any integer

- anonymous

Is there any trick to knowing which formula to try? I often grab one that would be allowed but isn't the one I should have used.

- anonymous

I only had to give solutions between 0 and 2pi but I like your further explaination

- anonymous

Oops. Explanation I mean

- phi

Unfortunately, with trig I think you just have to do a bunch of problems and build a sense of what might work.

- anonymous

Lol. I've done hundreds and am still picking the wrong ones. But I'm not giving up yet!

- anonymous

Thanks for help. my midterm is soon. Yikes.

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