## anonymous one year ago Will Medal! If 2tanx/ (1-tan^2x) = 1, then x can equal ____. Check all that apply. A) x=5pi/8 + npi B) x=pi/8 + npi C) x=3pi/8 + npi D) x=7pi/8 + npi

1. anonymous

I think it is A and B. But this is me taking a guess. I am terrible at trig. Sorry

2. welshfella

well first cross multiply to give 2 tanx = 1 - tan^2 x now rearrange this quadratic: tan^2 x + 2tanx - 1 = 0 can you factor this?

3. anonymous

Yes. I am thinking you factor to get tan^2x = -2tan(x)+1?

4. welshfella

no factor as you would any quadratic like x^2 + 3x + 2 = (x + 1)(x + 2)

5. welshfella

oh sorry - on second look this will not factor so you need to use the quadratic formula

6. anonymous

So would a = tan^2x b =2tanx and c=-1?

7. welshfella

no a , b and c are the coefficients of tan^2x and tan x so a = 1 , b = 2 and c = -1

8. anonymous

So x = +/- sqrt (2) in the end right?

9. anonymous

*-1 +/- sqrt(2)

10. welshfella

-2 +/- sqrt(4 - 4*1*-1) / 2 = -2 +/- sqrt8 / 2 = (-2 +/- 2 * sqrt2 )/ 2 = -1 +/- sqrt2 yep

11. anonymous

Awesome! So now do I just plug in this to my original equation?

12. welshfella

well they have the answer options in terms of pi so convert everything to decimals so for example -1 + sqrt2 = 0.4142 so find the angle in radians which is the tangent of 0.4142 and see if one of the options gives the the same angle

13. welshfella

eg angle with tangent 0.4142 = 0.39269 radians now check if pi/8 gives the same value

14. anonymous

Yes it does!

15. anonymous

So I'm assuming that is one

16. welshfella

ok so thats one result pi/8 + npi

17. welshfella

now try the other value for tan x

18. anonymous

-1 - sqrt 2 = -2.4142 degrees right?

19. welshfella

yes

20. anonymous

Ok so then the radians of -2.4142 is -0.042135

21. anonymous

I don't think I did it right

22. welshfella

no the -2.4142 is the value of tan x so you need tan-1(-2.4242)

23. anonymous

Oh ok that would make it -1.1781 which is basically the same as 3pi/8 right?

24. anonymous

So the answers would be pi/8 + npi and 3pi/8 + npi

25. welshfella

yes

26. anonymous

Great!! Thank you for taking the time to help me! I feel I really understand now! Thank you!

27. anonymous

Weird... I hit the sumit button and it told me it was 5pi/8 instead of 3/pi/8

28. welshfella

lets check that out

29. myininaya

$\frac{2tanx}{1-\tan^2x} = 1 \\ \tan(2x)=1$

30. myininaya

$\tan(u)=1 \text{ when } u=?$

31. welshfella

ahh i forgot that identity !

32. welshfella

u = pi/4 of course and x will be pi/8

33. welshfella

much easier that way lol!

34. myininaya

and there is also the u=5pi/4 so yep x=5pi/8

35. welshfella

yes

36. welshfella

-1.1781 is an incorrect result then? - an extraneous result right?

37. myininaya

does give tan(2x) approx 1 what number are you approximating ?

38. myininaya

oh -3pi/8 is what you are approximating

39. welshfella

sorry I dont follow you -yes

40. myininaya

that gives 1

41. myininaya

when you plug in

42. welshfella

and -3pi/8 is equivalent to 5pi/8

43. myininaya

is that not included in the solution: $\frac{5\pi}{8}+n \pi \\ \text{ if } n=-1 \text{ then } \frac{5\pi}{8}-\pi=\frac{-3\pi}{8}$ so I'm not sure why you say it is extraneous

44. welshfella

no - i was wrong its not extraneous

45. welshfella

still i think lehmad learned something along the line...

46. anonymous

Yes I learned a lot! Thank you guys for the help.