Results 1 to 4 of 4

Math Help - Unit circle and similar triangles

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    88

    Unit circle and similar triangles

    Consider the line y=t(x + 1), where t is rational, and passing through two points on the unit circle (around the origin O): the point P (1, 0) and the point R(x, y), say in the first quadrant. Consider the two angles given by RP and RO with the x-axis. Call them ψ and θ, respectively. Show that ψ = θ/2.

    So what I tried doing was subbing in t(x + 1)=y inso x^2+y^2=1,
    So after some long computations I get x=(1-t^2)/(1+t^2) and subbing in to get y=2t/(1+t^2)
    Now Im using SohCahToa to find the angles
    ψ andθ but Im not getting the right answer, My computations are correct, so am I doing something wrong?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Sep 2013
    From
    USA
    Posts
    314
    Thanks
    135

    Re: Unit circle and similar triangles

    Quote Originally Posted by calculuskid1 View Post
    Consider the line y=t(x + 1), where t is rational, and passing through two points on the unit circle (around the origin O): the point P (1, 0) and the point R(x, y), say in the first quadrant. Consider the two angles given by RP and RO with the x-axis. Call them ψ and θ, respectively. Show that ψ = θ/2.

    So what I tried doing was subbing in t(x + 1)=y inso x^2+y^2=1,
    So after some long computations I get x=(1-t^2)/(1+t^2) and subbing in to get y=2t/(1+t^2)
    Now Im using SohCahToa to find the angles
    ψ andθ but Im not getting the right answer, My computations are correct, so am I doing something wrong?
    Thanks

    The proof and the theore is valid for any angle θ
    Unit circle and similar triangles-untitled2.gif
    Last edited by votan; October 5th 2013 at 05:17 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2010
    Posts
    88

    Re: Unit circle and similar triangles

    Wow, that is MUCH simpler then everything I was trying.. Thank you for your help
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,930
    Thanks
    782

    Re: Unit circle and similar triangles

    Quote Originally Posted by calculuskid1 View Post
    Consider the line y=t(x + 1), where t is rational, and passing through two points on the unit circle (around the origin O): the point P (1, 0) and the point R(x, y), say in the first quadrant. Consider the two angles given by RP and RO with the x-axis. Call them ψ and θ, respectively. Show that ψ = θ/2.

    So what I tried doing was subbing in t(x + 1)=y inso x^2+y^2=1,
    So after some long computations I get x=(1-t^2)/(1+t^2) and subbing in to get y=2t/(1+t^2)
    Now Im using SohCahToa to find the angles
    ψ andθ but Im not getting the right answer, My computations are correct, so am I doing something wrong?
    Thanks
    If you want to solve it using your calculations: x=\dfrac{1-t^2}{1+t^2}, y = \dfrac{2t}{1+t^2}. So,
    \begin{align*}\tan(\psi) & = \dfrac{\mbox{opp}}{\mbox{adj}} \\ & = \dfrac{y}{1+x} \\ & = \dfrac{\tfrac{2t}{1+t^2}}{1 + \tfrac{1-t^2}{1+t^2}} \\ & = \dfrac{2t}{(1+t^2) + (1-t^2)} \\ & = t\end{align*}

    Next, we have
    \begin{align*}\tan(\theta) & = \dfrac{\mbox{opp}}{\mbox{adj}} \\ & = \dfrac{y}{x} \\ & = \dfrac{\tfrac{2t}{1+t^2}}{\tfrac{1-t^2}{1+t^2}} \\ & = \dfrac{2t}{1-t^2}\end{align*}

    Let's use the double angle formula for tangent to calculate \tan(2\psi) = \dfrac{2\tan(\psi)}{1-\tan^2(\psi)} = \dfrac{2t}{1-t^2} = \tan(\theta). Take arctan of both sides to get 2\psi = \theta.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Similar triangles
    Posted in the Geometry Forum
    Replies: 7
    Last Post: December 11th 2011, 08:47 PM
  2. similar triangles
    Posted in the Geometry Forum
    Replies: 5
    Last Post: November 19th 2011, 07:25 PM
  3. similar triangles...
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: October 6th 2011, 06:43 PM
  4. similar triangles
    Posted in the Geometry Forum
    Replies: 6
    Last Post: October 30th 2010, 02:58 PM
  5. similar triangles
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 21st 2009, 01:49 AM

Search Tags


/mathhelpforum @mathhelpforum