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Parametric and polar forms

Conic sections

Japanese school year: Math C

What you learn

Learn how to represent curves using parametric equations and describe coordinates using distances and angles in polar coordinate systems. These representations are particularly useful for analyzing complicated trajectories and rotationally symmetric physical systems. Prior proficiency with trigonometry, angle measurement, and vector operations will facilitate your understanding of these representations.

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Key points

This formula converts polar coordinates (r,θ)(r, \theta) (distance rr and angle heta heta) into Cartesian coordinates (x,y)(x, y). It is used when analyzing circular motion or rotational symmetry.

x=rcos⁡θ,y=rsin⁡θx = r\cos\theta, \quad y = r\sin\theta

These equations convert Cartesian coordinates (x,y)(x, y) into polar coordinates (r,θ)(r, \theta). Calculate the distance as r=x2+y2r = \sqrt{x^2 + y^2}, and choose θ\theta based on the quadrant of (x,y)(x, y).

r2=x2+y2,tan⁡θ=yxr^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x}

This parametric formula represents any point (x,y)(x, y) on the ellipse using a single parameter θ\theta. It simplifies optimization problems by reducing two variables to one trigonometric variable.

x=acos⁡θ,y=bsin⁡θx = a\cos\theta, \quad y = b\sin\theta

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