## Pianoman1996j Group Title How do you solve $\cos 2 x= cos x +2$ one year ago one year ago

1. phi Group Title

do you know cos(x+y) formula?

2. Pianoman1996j Group Title

Yes cosxcosy+sinxsiny

3. phi Group Title

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 Group Title

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 Group Title

does that make sense?

6. Pianoman1996j Group Title

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

7. phi Group Title

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 Group Title

*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 Group Title

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

10. Pianoman1996j Group Title

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 Group Title

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 Group Title

Of course! Thanks for helping me see.

13. phi Group Title

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 Group Title

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 Group Title

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

16. Pianoman1996j Group Title

Oops. Explanation I mean

17. phi Group Title

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

18. Pianoman1996j Group Title

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

19. Pianoman1996j Group Title

Thanks for help. my midterm is soon. Yikes.