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Parametric & Polar Equations Guide

1) Parametric equations define a curve using two equations of the form x=f(t) and y=g(t), where t is a parameter. Eliminating t gives the cartesian equation of the curve. 2) Polar coordinates represent a point in the plane using its distance r from the origin and its angle θ from the positive x-axis. 3) Polar equations relate r and θ, and can be converted to cartesian equations by using the relationships x=r cosθ and y=r sinθ.

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Loraine Ruth Lumen
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113 views14 pages

Parametric & Polar Equations Guide

1) Parametric equations define a curve using two equations of the form x=f(t) and y=g(t), where t is a parameter. Eliminating t gives the cartesian equation of the curve. 2) Polar coordinates represent a point in the plane using its distance r from the origin and its angle θ from the positive x-axis. 3) Polar equations relate r and θ, and can be converted to cartesian equations by using the relationships x=r cosθ and y=r sinθ.

Uploaded by

Loraine Ruth Lumen
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Parametric Equations

Polar Coordinates

PARAMETRIC EQUATIONS AND


POLAR COORDINATES
Parametric Equations
Polar Coordinates

1 Parametric equations and plane curves


2 Polar coordinates
3 Polar equations and curves
Parametric Equations
Polar Coordinates
Parametric Equations
Polar Coordinates

Definition

Parametric Equations
A curve given by

x = f (t) and y = g(t)

is called a parametrically defined curve, or simply, a plane


curve. The functions

x = f (t) and y = g(t)

are called parametric equations of the curve and the variable t


is called the parameter.
Parametric Equations
Polar Coordinates

Eliminating the Parameter t

If the parameter t is eliminated from the pair of equations

x = f (t) and y = g(t),

we obtain the cartesian equation of the curve.


Parametric Equations
Polar Coordinates

Examples

Example 1
Find a cartesian equation of the curve defined by the
parametric equations

x = 2t − 3 and y = 4t − 1.

Example 2
What curve is represented by the parametric equations

x = 3 cos t and y = 3 sin t,

0 ≤ t ≤ 2π?
Parametric Equations
Polar Coordinates

Polar Coordinates
Parametric Equations
Polar Coordinates

1 θ > 0 if measured counterclockwise


2 θ < 0 if measured clockwise

Example
Plot the following points(given in polar coordinates).
2, π6

1

2, −11π

2
6
3 −2, π6
2, 7π

4
6 
5 2, 13π
6
Parametric Equations
Polar Coordinates
Parametric Equations
Polar Coordinates

Remarks
1 The angle associated with a given point is not unique.

2 (r, θ) = (r, θ + 2π)


Parametric Equations
Polar Coordinates

Relating Polar and Cartesian Coordinates


Polar Equation
An equation in polar coordinates is called a polar equation.

x = r cos θ ; y = r sin θ ; r2 = x2 + y 2
Parametric Equations
Polar Coordinates

Polar Equation to Cartesian Equation

Replace the following polar equations by equivalent Cartesian


equations, and identify their graphs.
1 r cos θ = 2
2 r2 = 4r cos θ
Parametric Equations
Polar Coordinates

Cartesian Equation to Polar Equation

1 Find the equation of the line y = 3x + 2 in polar


coordinates.
2 Find a polar equation for the circle x2 + (y − 3)2 = 9.
Parametric Equations
Polar Coordinates

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