Proving Rotation of Conic Using Polar Equations: Am I on the Right Track?

  • Thread starter ballahboy
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In summary, The conversation discusses the process of replacing x and y in a polar equation with new variables x' and y', using a geometric drawing as a starting point. The conversation also mentions using difference identities and clarifies the meanings of rsin(alpha) and rcos(alpha). The speaker is unsure if their approach is correct but notes that rsin(theta) is equivalent to y and rcos(theta) is equivalent to x. Overall, the conversation aims to prove a mathematical concept using polar equations instead of matrices.
  • #1
ballahboy
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Ok. So i got that part where you replace x with x'cos(theta)+y'sin(theta) and y with -x'sin(theta)+y'cos(theta). My book shows us the proof for this using matrices. What my teacher wants us to do its prove this using polar equations or something like that.

I started out with the drawing attached(teacher told us to use it). From that i got x'=rcos(theta-alpha) and y'=rsin(theta-alpha). Using the difference identities, i got x'=rcos(theta)cos(alpha)+rsin(theta)sin(alpha) and y'=rsin(theta)cos(alpha)-rcos(theta)sin(alpha). I kinda got stuck after this.. Am i on the right track at all? or is it completely off
 

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  • #2
Line up the corresponding equations.

What is rsin(alpha)? and rcos(alpha)?
 
  • #3
robphy said:
What is rsin(alpha)? and rcos(alpha)?

umm.. iono about that but I know that rsin(theta) is y and rcos(theta) is x.. would that be the same?
 

Related to Proving Rotation of Conic Using Polar Equations: Am I on the Right Track?

1. What is the rotation of a conic?

The rotation of a conic is a transformation that involves changing the orientation of a conic section, such as a circle, ellipse, parabola, or hyperbola. This transformation is achieved by rotating the axes of the conic section by a certain angle.

2. How is the rotation of a conic performed?

The rotation of a conic is performed by using a rotation matrix, which is a mathematical tool that describes how to rotate points in a plane by a given angle. The matrix is multiplied by the coordinates of the conic section to obtain the rotated coordinates.

3. What is the equation for the rotation of a conic?

The equation for the rotation of a conic is given by x' = x*cosθ - y*sinθ and y' = x*sinθ + y*cosθ, where (x,y) are the original coordinates of the conic and (x',y') are the rotated coordinates. θ represents the angle of rotation.

4. What effect does rotation have on the shape of a conic?

The effect of rotation on the shape of a conic depends on the angle of rotation. If the angle is 0° or 180°, there is no change in shape. However, if the angle is between 0° and 90°, the conic becomes more elongated or stretched out. If the angle is between 90° and 180°, the conic becomes more compressed or squished.

5. What real-world applications use the concept of rotation of a conic?

The concept of rotation of a conic has various applications in fields such as engineering, physics, and astronomy. For example, it is used to model the orbits of planets around the sun, the rotation of satellites in space, and the motion of objects in projectile motion. It is also used in computer graphics and animation to create rotating objects and shapes.

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