A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

Solve Trig Equation \( tan \theta = \frac{2\sqrt{3}}{3} sin \theta \)

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I need to get the solution sets

  2. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hint u can write tan(theta) as sin(theta)/cos(theta)

  3. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Well, first off we should note that we cannot have solutions that would make cosine equal to 0, as then we would have something undefined. So let me change tangent into sin/cos and do this: \[\frac{ \sin\theta }{ \cos\theta } = \frac{ 2\sqrt{3} }{ 3 }\sin\theta\] \[\sin\theta = \frac{ 2\sqrt{3} }{ 3 }\sin\theta\cos\theta\] \[\frac{ 2\sqrt{3} }{ 3 }\sin\theta\cos\theta - \sin\theta = 0\] Now factor and see if you solve from there

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433617958179:dw|

  5. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    nw cancel sin(theta) from both sides of the equation

  6. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    then u'll b left with 1/cos(theta) = 2/root(3) cross multiplying we get cos(theta) = root(3)/2 which is equal to cos(30) therefore theta = 30degrees

  7. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    In general, you do not want to cancel sin(theta) from both sides of the equation, or divide by any variable expression for that manner. If you do this, you could potentially lose solutions.

  8. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    cos(theta) = cos(30) = cos(330)

  9. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    To show this, by inspection you can see that 0 would be a solution to this equation. Yet if you divide out sin(theta), you completely lose 0 as a solution in what would be remaining.

  10. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am stuck on factoring the last part

  11. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\tan(0) = \frac{ 2\sqrt{3} }{ 3 }\sin(0)\] YOu get 0 = 0. You lose this solution if you cancel out sin(theta) from both sides.

  12. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok m posting a full solution

  13. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    As for the factoring, where I have \[\frac{ 2\sqrt{3} }{ 3 }\sin\theta\cos\theta - \sin\theta = 0\] you can factor out sin(theta) \[\sin \theta(\frac{ 2\sqrt{3} }{ 3 }\cos\theta -1) = 0\] This gives two possibilities. \[\sin\theta = 0\] or \[\frac{ 2\sqrt{3} }{ 3 }\cos\theta -1 = 0\]

  14. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433618602641:dw|

  15. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You cannot divide by sin(theta). In general, you cannot divide by variable expression. Notice that I had one of my potential results when I factored as sin(theta) = 0. You completely lose this solution if you divide out sin(theta), you are not allowed to do this @Nixy

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Where does the -1 come from? Is it because cos -sin = cos - 1 sin ???

  17. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    As a small example of why the -1, consider 2x-2. I can factor a 2 out of this expression and get 2(x-1). When you factor a number out of itself, it leaves behind a 1. You can also distribute to check this result if you wish.

  18. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok I see thanks

  19. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    But from above, I have solutions that will come from \[\sin\theta = 0\] and \[\frac{ 2\sqrt{3} }{ 3 }\cos\theta - 1 = 0\]

  20. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    √3 tan x = 2 sin x √3 tan x - 2 sin x = 0 sin x ( √3/cos x - 2 ) = 0 => sin x = 0 or √3 / cos x - 2 = 0 when sin x = 0 x = 2kpi where k= 0,1,2,3......... when √3 / cos x - 2 = 0 => cos x = √3/2

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I think I got it. Thanks to you all

  22. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You're welcome :)

  23. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yep I got it. My solutions were 0, pi, 5pi/6, 7pi/6 Thanks once again to you all.

  24. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    :)

  25. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.