Results 1 to 11 of 11

Math Help - Find the value R

  1. #1
    Junior Member
    Joined
    Aug 2012
    From
    The Earth
    Posts
    71
    Thanks
    1

    Find the value R

    Given that the two equations:
    R sin⁡〖60〗=10 sin⁡〖θ〗+v
    R cos⁡〖60〗=10 cos⁡〖θ〗

    How do I get the value R ? Please show me how to do it. Thanks.
    Last edited by alexander9408; September 12th 2012 at 10:28 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member kalyanram's Avatar
    Joined
    Jun 2008
    From
    Bangalore, India
    Posts
    143
    Thanks
    14

    Re: Find the value R

    Is \theta \hspace{2mm}or \hspace{2mm}v known.
    Last edited by kalyanram; September 12th 2012 at 10:34 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member kalyanram's Avatar
    Joined
    Jun 2008
    From
    Bangalore, India
    Posts
    143
    Thanks
    14

    Re: Find the value R

    Quote Originally Posted by alexander9408 View Post
    Given that the two equations:
    R sin⁡〖60〗=10 sin⁡〖θ〗+v
    R cos⁡〖60〗=10 cos⁡〖θ〗

    How do I get the value R ? Please show me how to do it. Thanks.
    I assume v is known.

    \left( \frac{\sqrt{3}}{2} R - v}\right)^2 + \frac{R^2}{4} = 100

    This can be solved for R=\frac{\sqrt{3}v \pm \sqrt{400-v^2}}{2}
    Last edited by kalyanram; September 12th 2012 at 10:42 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Aug 2012
    From
    The Earth
    Posts
    71
    Thanks
    1

    Re: Find the value R

    both are unknown
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member kalyanram's Avatar
    Joined
    Jun 2008
    From
    Bangalore, India
    Posts
    143
    Thanks
    14

    Re: Find the value R

    Well then there are three unknowns R, \theta, v and two equations. This cannot be solved for a unique solution.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Aug 2012
    From
    The Earth
    Posts
    71
    Thanks
    1

    Re: Find the value R

    well then, can you form this equation θ=60 - sin^(-1)⁡〖v/20〗using the two equations above?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member kalyanram's Avatar
    Joined
    Jun 2008
    From
    Bangalore, India
    Posts
    143
    Thanks
    14

    Re: Find the value R

    Quote Originally Posted by alexander9408 View Post
    well then, can you form this equation θ=60 - sin^(-1)⁡〖v/20〗using the two equations above?
    Yes this can be done by eliminating R from the above two equations?
    Spoiler:

    Dividing the given equations we have
    \sqrt{3} = \frac{10 \sin \theta + v}{10 \cos \theta} \implies 10\sqrt{3} \cos \theta - 10 \sin \theta = v
    \implies 20 \left( \frac{\sqrt{3}}{2} \cos \theta - \frac{1}{2} \sin \theta \right) = v
    \implies \sin \left( 60^{o} - \theta \right) = \frac{v}{20}
    \implies \theta = 60^{o} - \sin^{-1} \left( \frac{v}{20} \right)

    NOTE: Before division by 10 \cos \theta you need to argue that you are not dividing by zero first.
    Last edited by kalyanram; September 12th 2012 at 12:36 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Aug 2012
    From
    The Earth
    Posts
    71
    Thanks
    1

    Re: Find the value R

    Quote Originally Posted by kalyanram View Post

    NOTE: Before division by 10 \cos \theta you need to argue that you are not dividing by zero first.
    [/SPOILER]
    What do you mean by this?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member kalyanram's Avatar
    Joined
    Jun 2008
    From
    Bangalore, India
    Posts
    143
    Thanks
    14

    Re: Find the value R

    Quote Originally Posted by alexander9408 View Post
    What do you mean by this?
    Have you considered the case when \theta=\frac{\pi}{2}.

    By the way the general solution is \theta = n\pi + (-1)^n \left( \frac{\pi}{3} - \sin^{-1} \left( \frac{v}{20}\right) \right), n \in \mathbb{Z}. (why?)
    Last edited by kalyanram; September 12th 2012 at 01:04 PM.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    Aug 2012
    From
    The Earth
    Posts
    71
    Thanks
    1

    Re: Find the value R

    Quote Originally Posted by kalyanram View Post
    Have you considered the case when \theta=\frac{\pi}{2}.

    By the way the general solution is \theta = n\pi + (-1)^n \left( \frac{\pi}{3} - \sin^{-1} \left( \frac{v}{20}\right) \right), n \in \mathbb{Z}. (why?)
    Since the question have been given that the v<10 , when I substitute v=10 to the equation, then I will get θ = 30 , so the θ won't exceed 30.
    The value of sin is positive at first and second quadrant, and negative at third and fourth quadrant, is that so?
    Last edited by alexander9408; September 12th 2012 at 01:25 PM.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member kalyanram's Avatar
    Joined
    Jun 2008
    From
    Bangalore, India
    Posts
    143
    Thanks
    14

    Re: Find the value R

    Quote Originally Posted by alexander9408 View Post
    Since the question have been given that the v<10
    When \theta = \frac{\pi}{2} we have R=0 and v=-10 which satisfies the condition v<10.

    Anyway the point I wanted to highlight was not that. I am probably confusing you more that I intend to so here is how I would like you to analyze.

    Step1: There are three unknown R,\theta, v, and two equations so if you want to relate any two you have to eliminate the other.
    Relating R \& v by eliminating \theta was shown above. Now here the discriminant of the quadratic (400 - v^2) should be > 0 for the range of values specified on v this gives us an additional criteria that |v| \le 20. So permissible values of v are -20 \le v < 10.

    Step 2: Relating \theta and v
    While eliminating any variable by trying to divide two expressions (like I did) you have to be careful as to the expression in the denominator is \ne 0. This is the case when \theta = \frac{\pi}{2}. So you can deal with it as a separate case and argue it out, or try to avoid division if possible. And in this case you can avoid division of the two expressions as shown below:
    From the first equation w
    We have R = \frac{2}{\sqrt{3}} \left( 10 \sin \theta + v \right), R = 20 \cos \theta

    20 \cos \theta = \frac{2}{\sqrt{3}} \left( 10 \sin \theta + v \right) from here we again get

    10 \sqrt{3} \cos \theta - 10 \sin \theta = v which has been solved above. The point to be noted here is that eliminating R as shown here we have avoided any possibility of division by 0.

    Step 3: General values of \theta.

    Quote Originally Posted by alexander9408 View Post
    when I substitute v=10 to the equation, then I will get θ = 30 , so the θ won't exceed 30. The value of sin is positive at first and second quadrant, and negative at third and fourth quadrant, is that so?
    As I have mentioned above when you obtain a solution for a variable it is all the more important to verify if the solution is valid. When solving for Step 2, we obtain \theta = \frac{\pi}{3} - sin^{-1} \left( \frac{v}{20} \right) as the principle values of the angle. As stated by you we already know that \sin (\pi - \theta) = \sin \theta and also \sin (2n\pi + \theta) = \sin \theta, \forall n \in \mathbb{Z} so we need all possible solutions of \theta we obtain the form \theta = n\pi + (-1)^n \left( \frac{\pi}{3} - sin^{-1} \left( \frac{v}{20}\right)\right). Now we need to see if this solution is a valid solution for that we need |v|< 20(why?) which is also consistent with what we got in the solution for R.
    Last edited by kalyanram; September 12th 2012 at 11:17 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove The Equation
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: May 29th 2010, 07:57 PM
  2. prove lim equation
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: December 4th 2009, 02:41 AM
  3. Prove equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 18th 2009, 05:08 PM
  4. Replies: 3
    Last Post: January 26th 2009, 12:19 PM
  5. Prove that the equation...?
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: July 27th 2008, 04:50 AM

/mathhelpforum @mathhelpforum