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Parametric Equations

Parametric equations define dependent variables x and y in terms of an independent parameter t, allowing for the representation of curves such as circles, parabolas, ellipses, and projectile motion. The general form is x = f(t) and y = g(t), with t often representing time. To convert parametric equations to Cartesian form, the parameter t can be eliminated, as shown in the example x = cos(t), y = sin(t) leading to x^2 + y^2 = 1.

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17 views2 pages

Parametric Equations

Parametric equations define dependent variables x and y in terms of an independent parameter t, allowing for the representation of curves such as circles, parabolas, ellipses, and projectile motion. The general form is x = f(t) and y = g(t), with t often representing time. To convert parametric equations to Cartesian form, the parameter t can be eliminated, as shown in the example x = cos(t), y = sin(t) leading to x^2 + y^2 = 1.

Uploaded by

yofop81896
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:

Parametric equations express a set of related quantities as functions of an independent


parameter (often denoted as tt).

Instead of writing yy as a function of xx, we define both xx and yy in terms of a third variable
(parameter) tt:

General Form:
x=f(t),y=g(t)x = f(t), \quad y = g(t)

Where:

 xx and yy are dependent variables


 tt is the parameter (usually represents time)

Examples:

1. Circle (radius rr)


x=rcos⁡(t),y=rsin⁡(t),0≤t<2πx = r \cos(t), \quad y = r \sin(t), \quad 0 \le t < 2\pi
2. Parabola
x=t,y=t2⇒Eliminating t⇒y=x2x = t, \quad y = t^2 \quad \Rightarrow \text{Eliminating } t
\Rightarrow y = x^2
3. Ellipse
x=acos⁡(t),y=bsin⁡(t),0≤t<2πx = a \cos(t), \quad y = b \sin(t), \quad 0 \le t < 2\pi
4. Projectile Motion
x=v0cos⁡(θ)t,y=v0sin⁡(θ)t−12gt2x = v_0 \cos(\theta) t, \quad y = v_0 \sin(\theta) t -
\frac{1}{2}gt^2

To Convert to Cartesian Form:

Eliminate tt from the parametric equations.

Example:

x=cos⁡(t),y=sin⁡(t)⇒x2+y2=1x = \cos(t), \quad y = \sin(t) \Rightarrow x^2 + y^2 = 1

Let me know if you want graphs, applications, or 3D parametric forms too.

Parametric Equations:
Parametric equations express a set of related quantities as functions of an independent
parameter (often denoted as tt).

Instead of writing yy as a function of xx, we define both xx and yy in terms of a third variable
(parameter) tt:

General Form:

x=f(t),y=g(t)x = f(t), \quad y = g(t)

Where:

 xx and yy are dependent variables


 tt is the parameter (usually represents time)

Examples:

1. Circle (radius rr)


x=rcos⁡(t),y=rsin⁡(t),0≤t<2πx = r \cos(t), \quad y = r \sin(t), \quad 0 \le t < 2\pi
2. Parabola
x=t,y=t2⇒Eliminating t⇒y=x2x = t, \quad y = t^2 \quad \Rightarrow \text{Eliminating } t
\Rightarrow y = x^2
3. Ellipse
x=acos⁡(t),y=bsin⁡(t),0≤t<2πx = a \cos(t), \quad y = b \sin(t), \quad 0 \le t < 2\pi
4. Projectile Motion
x=v0cos⁡(θ)t,y=v0sin⁡(θ)t−12gt2x = v_0 \cos(\theta) t, \quad y = v_0 \sin(\theta) t -
\frac{1}{2}gt^2

To Convert to Cartesian Form:

Eliminate tt from the parametric equations.

Example:

x=cos⁡(t),y=sin⁡(t)⇒x2+y2=1x = \cos(t), \quad y = \sin(t) \Rightarrow x^2 + y^2 = 1

Let me know if you want graphs, applications, or 3D parametric forms too.

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