## onegirl Group Title 2 cos^2 x - 3 cos x + 1 = 0 for 0 is less than or equal to x is less than 2pi one year ago one year ago

1. dpaInc Group Title

the first term with the cosine... is it: $$\large 2cos^2x$$ or $$\large 2cos(2x)$$ i'm thinking it's the first one?

2. onegirl Group Title

yea the first one

3. dpaInc Group Title

think of it this way.. let y=cosx so your equation becomes: $$\large 2y^2-3y+1=0$$ can you solve this quadratic?

4. onegirl Group Title

yea i can

5. dpaInc Group Title

what is/are the solutions ? y = ???

6. z3529080 Group Title

cos2x=2cos^2(x)-1

7. z3529080 Group Title

2(2cos^2(x)-1)-3cosx+1=0 -> 4cos^2(x)-3cosx-1=0 -> (4cosx+1)(cosx-1)=0

8. onegirl Group Title

x1 = -0.5 x2 = -1

9. z3529080 Group Title

so set 4cosx+1=0 or cos-1 =0

10. dpaInc Group Title

hmmm.. i got the same but POSITIVE values... i'm double checking...

11. dpaInc Group Title

yeah... they should be positive: $$\large 2y^2-3y+1=0$$ $$\large (2y-1)(y-1)=0$$ so 2y - 1= 0 gives y=1/2 y-1 = 0 gives y=1

12. dpaInc Group Title

ok so far?

13. onegirl Group Title

yea

14. z3529080 Group Title

you forgot to x2

15. z3529080 Group Title

it is 2cos2x

16. onegirl Group Title

it is 2 cos^2 x - 3

17. z3529080 Group Title

oh i thought it is cos2x

18. dpaInc Group Title

since y=cosx, we have these two equations: $$\large cosx=\frac{1}{2}$$ and $$\large cosx=-1$$ can you solve these?

19. dpaInc Group Title

oops... that second one should be POSITIVE one...

20. dpaInc Group Title

do you use the unit circle?

21. onegirl Group Title

yes i do

22. joshi Group Title

this is probably a stupid question, but i'm a bit confused. you said let y = cosx so what happens to the first term 2cos2x? like what about the 2 infront of the x?

23. onegirl Group Title

@joshi its 2 cos^2 x -3 i made a mistake in typing it

24. dpaInc Group Title

ok... so look at what angle gives you a cosine of 1/2.... HINT: there are two of 'em from 0 to 2pi

25. joshi Group Title

ohhh ok that makes more sense haha

26. dpaInc Group Title

yeah... i got a clarification from the asker with my first question... :)

27. onegirl Group Title

60 degrees?

28. dpaInc Group Title

|dw:1350069794735:dw| yep... 60 degrees is one of 'em....

29. onegirl Group Title

or pi/3

30. dpaInc Group Title

you got the second angle?

31. onegirl Group Title

and 45 degrees?

32. dpaInc Group Title

nope... we're still solving cosx = 1/2 , right? x=60 degrees or pi/3 is one... the second solution is in the fourth quadrant...

33. dpaInc Group Title

|dw:1350070145499:dw|

34. dpaInc Group Title

or take a look at page 3: http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

35. onegirl Group Title

240 degrees?

36. dpaInc Group Title

240 degrees is in the third quadrant... this is the fourth quadrant: |dw:1350070347242:dw|

37. onegirl Group Title

ohh ok 300 degrees..

38. dpaInc Group Title

yes... :) cos(60 degrees) = 1/2 and cos(300 degrees) = 1/2 now solve the other equation: cos x = 1

39. dpaInc Group Title

and remember, $$\large 0\color {red}{\le} x<2\pi$$

40. dpaInc Group Title

there should be only one solution for this one....

41. dpaInc Group Title

using the unit circle, what angle gives you an x-coordinate of 1?

42. onegirl Group Title

360 degrees?

43. dpaInc Group Title

yes but 2pi is not included in the interval you're looking at.. 360 degrees = 2pi so what other angle gives you an x-coordinate of 1?

44. dpaInc Group Title

HINT: $$\huge \le$$

45. onegirl Group Title

ohhh yea it will 180 degrees

46. dpaInc Group Title

no.. cos(180 degrees) = -1 try again....

47. dpaInc Group Title

cos x = 1 where $$\large 0\color {red}{\le} x<2\pi$$

48. onegirl Group Title

3pi/2?

49. dpaInc Group Title

what about the left side of $$\large 0\color {red}{\le} x<2\pi$$ ???? x = 0 ? cos 0 = ???

50. dpaInc Group Title

ok... let's put it this way... what's an angle that is coterminal with 360 degrees?

51. onegirl Group Title

positive 360 and negative -360

52. dpaInc Group Title

no.. the angle you want is 0 because cos(0 degrees) = 1 so you have 3 solutions: x=0, 60, 300 degrees but since the problem was stated in terms of radians, i'd convert those angles.

53. onegirl Group Title

so it will be x=1.823 radian