## anonymous one year ago Use a graphing calculator to solve the equation -3 cos t = 1 in the interval from

1. anonymous

$0\le \theta \le 2x$

2. anonymous

Round to the nearest hundredth and show your work.

3. freckles

is x suppose to be pi?

4. anonymous

oh yeah sorry it is

5. freckles

$\cos(t)=\frac{-1}{3} \text{ on } 0 \le t \le 2\pi \\ \text{ one solution can be found just by taking } \arccos( ) \text{ of both sides } \\ \text{ other one we can find \it by using that } \\ \text{ cosine is even and has period } 2 \pi \\ \cos(-t)=\cos(t) \\ \text{ so we have } \\ \cos(-t+2\pi)=\frac{-1}{3} \\ \text{ take } \arccos( ) \text{ of both sides and solve for } t$

6. anonymous

@freckles can you keep going cuz i really dont know how to do this

7. anonymous

I fanned you and gave you a medal

8. freckles

well were you not able to the first solution? it is just taking arccos( ) of both sides

9. freckles

$\cos(t)=\frac{-1}{3} \text{ we could say one solution is } t=\arccos(\frac{-1}{3})$

10. anonymous

No im really confused, that's why i like when people go step by step

11. freckles

Oh I thought I gave you step by step

12. freckles

all I did there was take arccos( ) of both sides for that one solution

13. freckles

just as I said to do in the steps above

14. anonymous

you did, but is that the answer?

15. freckles

that is one solution i also said how to find the other

16. freckles

use the fact that cos is even and has period 2pi

17. anonymous

18. freckles

how did you get that?

19. anonymous

No i'm asking you, from your step by step that's what is above that everything is equalling -1/3

20. freckles

the equation is cos(t)=-1/3 we aren't solving for cos(t) if we were then we would done and the answer is -1/3 we are solving for t

21. freckles

one solution is given by taking arccos( ) of both sides of the equation cos(t)=-1/3

22. anonymous

ok how do we do that?

23. freckles

one solution is given by t=arccos(-1/3)

24. freckles

and then the other solution is given by using the fact that cos is even and has period 2pi

25. anonymous

t=4.37255207±2πn

26. freckles

let me give you an easy example to follow: $\cos(\theta)=\frac{-1}{2} , \text{ where } 0 \le \theta \le 2 \pi \\ \text{ we know we are suppose to get solutions } \theta=\frac{2 \pi }{3} ,\frac{4\pi}{3} \\ \text{ we see we can get these solutions by doing : } \\ \text{ Answer one given by just takinng } \arccos( ) \text{ of both sides } \\ \theta=\arccos(\frac{-1}{2}) \\ \\ \text{ now the other solution comes from using the fact that } \\ \text{ cosine is even function and has period } 2\pi \\ \cos(-\theta+2\pi)=\frac{-1}{2} \\ \text{ solve for } \theta \text{ by first taking } \arccos( ) \text{ of both sides } \\ -\theta+2\pi=\arccos(\frac{-1}{2}) \\ \text{ now subtract } 2\pi \text{ on both sides } \\ -\theta=\arccos(\frac{-1}{2})-2\pi \\ \text{ last step multiply -1 on both sides } \\ \theta=-\arccos(\frac{-1}{2})+2\pi \\ \\ \text{ so the solutions to } \cos(\theta)=\frac{-1}{2} \text{ on } 0 \le \theta \le 2\pi \text{ are } \\ \theta=\arccos(\frac{-1}{2}) \text{ or } \theta=-\arccos(\frac{-1}{2})+2\pi \\ \text{ we can simplify these solutions } \\ \text{ We know } \arccos(\frac{-1}{2})=\frac{2\pi}{3} \\ \\\ \text{ so we have } \theta=\frac{2\pi}{3} \text{ or } \theta=-\frac{2\pi}{3}+2\pi=\frac{4\pi}{3} \\ \text{ just as we got way above just from using the unit circle }$

27. freckles

|dw:1437253274947:dw|

28. freckles

anyways you can use the exact same process I used above

29. freckles

just take arccos() of both sides to get one solution

30. freckles

then rewrite the equation as cos(-t+2pi)=whatever where whatever is between -1 and 1 (inclusive) and take arccos( ) of both sides and then solve that resulting equation for t

31. freckles

$\cos(t)=\frac{-1}{3} \\ \text{ one solution is } t=\arccos(\frac{-1}{3} ) \\ \text{ the other solution comes from solving } \cos(-t+2\pi)=\frac{-1}{3}$

32. freckles

if I take arccos( ) of both sides of your second equation -t+2pi=arccos(-1/3) can you try to solve this for t?

33. freckles

it is just two steps

34. anonymous

35. freckles

do you know how to solve -x+4=6 for x?

36. anonymous

yes

37. freckles

like you would subtract 4 on both sides -x=6-4 -x=2 and last step is multiply -1 on both sides x=-2 like this is the same type of equation here: $-t+2\pi=\arccos(\frac{-1}{3} ) \\ \text{ subtract } 2\pi \text{ on both sides } \\ -t =\arccos(\frac{-1}{3})-2\pi \\ \text{ multiply -1 on both sides } \\ t=-\arccos(\frac{-1}{3})+2\pi$

38. freckles

you have both solutions you can use your calculator to approximate them if you want

39. freckles

I think the instructions above do say to round to nearest hundredth so you will have to break out the calculator

40. anonymous

Will you help me to the final answer?

41. freckles

I gave you both exact solutions... $t=\arccos(\frac{-1}{3}) \\ t=-\arccos(\frac{-1}{3})+2\pi$ ?

42. freckles

oh are you saying you don't know how to plug it in the calculator?

43. freckles

there should be arcos typed on your calculator right above the cos button

44. freckles

arccos*

45. freckles

are it could be written as cos^(-1)

46. freckles

do you see it?

47. freckles

to access it on some calculators you have to press (shift or 2nd) can't remember which one

48. anonymous

No i cant get a good answer it keeps saying error

49. anonymous

t=8.19381854

50. freckles

what calculator do you use i can try to look it up and tell you what buttons to push

51. anonymous

t=8.19381854

52. anonymous

is that right?

53. freckles

oh i was waiting on you to tell me the calculator type but no

54. anonymous

It's on a website cuz i dont have a calculator

55. freckles

arccos(-1/3)=1.91 by using a calculator so we already have that as a solution now replace the arccos(-1/3) in the other solution with 1.91 $-1.91+2\pi$ you can do 3.14 as an approximation for pi 3.14*2=6.28 so what is -1.91+6.28

56. anonymous

4.37

57. freckles

yes so remember you have two answers

58. anonymous

So 4.37 is one, what's the other?

59. freckles

:(

60. anonymous

what?

61. freckles

I just feel like you haven't listened to me $t=\arccos(\frac{-1}{3}) \approx 1.91 \\ t=-\arccos(\frac{-1}{3})+2\pi \approx -1.91+2(3.14) =4.37$

62. anonymous

No I have, im just learning. Im sorry

63. freckles

ok well I hope you understood some of what I said

64. freckles

I have to leave now any questions before I go

65. anonymous

Im gonna overview it right now, thank you very much for your help i appreciate it!