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Math Help - Proving the Standard Counter-Clockwise Rotation Matrix and more.

  1. #1
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    Proving the Standard Counter-Clockwise Rotation Matrix and more.

    I'm having trouble trying to prove this question.
    Show that the linear transformation y = Ax where
    A=<br />
\[ \left[ \begin{array}{cc}<br />
  \cos\theta& -\sin\theta  \\<br />
  \sin\theta& \cos\theta    \end{array} \right]\], x = \[\left[\begin{array}{c}x_1\\x_2\end{array}\right]\] and y = \[\left[\begin{array}{c}y_1\\y_2\end{array}\right]\] is a counterclockwise rotation of the Cartesian x_1x_2-coordinate system in the plane about the origin, where \theta is the angle of rotation.

    Here's my problem. The way I was attempting to prove this problem was to use the knowledge the relationship between polar and Cartesian xy-plane. I.e. (x,y)=(r\cos\theta,r\sin\theta) that was going just fine until I needed to use the trig identities that says \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta the problem I have is that the question immediately following the one above asks.
    Use the above result to prove the following trigonometric identities:
    \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
    \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\  sin\beta

    Now I'm assuming my professor wants me to prove the rotation matrix without using trig identities. This is where I because all my proofs and arguments seem way to vague or rely on past knowledge of the cartesian vs polar systems. This problem was designed to be easy, so why am I having such a hard time? So if anyone has ideas to prove the top part that would be great. All the proofs I have found on the internet rely on trig identities(like mine) or just give the rotation matrix without any justification for it. I.e my linear algebra books just gives the rotation matrix without saying why it is that. Well thanks for taking the time to read this.
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  2. #2
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    Quote Originally Posted by fizzle45 View Post
    I'm having trouble trying to prove this question.
    Show that the linear transformation y = Ax where
    A=<br />
\[ \left[ \begin{array}{cc}<br />
  \cos\theta& -\sin\theta  \\<br />
  \sin\theta& \cos\theta    \end{array} \right]\], x = \[\left[\begin{array}{c}x_1\\x_2\end{array}\right]\] and y = \[\left[\begin{array}{c}y_1\\y_2\end{array}\right]\] is a counterclockwise rotation of the Cartesian x_1x_2-coordinate system in the plane about the origin, where \theta is the angle of rotation.

    Here's my problem. The way I was attempting to prove this problem was to use the knowledge the relationship between polar and Cartesian xy-plane. I.e. (x,y)=(r\cos\theta,r\sin\theta) that was going just fine until I needed to use the trig identities that says \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta the problem I have is that the question immediately following the one above asks.
    Use the above result to prove the following trigonometric identities:
    \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
    \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\  sin\beta

    Now I'm assuming my professor wants me to prove the rotation matrix without using trig identities.

    Perhaps he does, but (1)I can't understand why (s)he'd want such a thing, and (2) I'm not sure whether it is possible

    to prove such a thing without trigonometry, even more when you say below that the problem is suposed to be easy, so what's

    easier than trigonometry applied to this problem?

    Tonio



    This is where I because all my proofs and arguments seem way to vague or rely on past knowledge of the cartesian vs polar systems. This problem was designed to be easy, so why am I having such a hard time? So if anyone has ideas to prove the top part that would be great. All the proofs I have found on the internet rely on trig identities(like mine) or just give the rotation matrix without any justification for it. I.e my linear algebra books just gives the rotation matrix without saying why it is that. Well thanks for taking the time to read this.
    .
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