Pianoman1996j 2 years ago How do you solve $\cos 2 x= cos x +2$

1. phi

do you know cos(x+y) formula?

2. Pianoman1996j

Yes cosxcosy+sinxsiny

3. 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

4. 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

5. phi

does that make sense?

6. Pianoman1996j

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

7. 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

8. phi

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

9. Pianoman1996j

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

10. Pianoman1996j

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?

11. 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

12. Pianoman1996j

Of course! Thanks for helping me see.

13. 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

14. Pianoman1996j

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.

15. Pianoman1996j

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

16. Pianoman1996j

Oops. Explanation I mean

17. phi

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

18. Pianoman1996j

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

19. Pianoman1996j

Thanks for help. my midterm is soon. Yikes.