10. Parametric Equations

Writing Parametric Equations

10. Parametric Equations

Writing Parametric Equations

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Multiple Choice

Write parametric equations for the rectangular equation below.
$x^2+y^2=25$

$x=25\sin t$ ; $y=25\cos t$

$x=25\cos t$ ; $y=25\sin t$

$x=5\sin t$ ; $y=5\cos t$

$x=5\cos t$ ; $y=5\sin t$

Verified step by step guidance

Start by recognizing that the given rectangular equation $x^2 + y^2 = 25$ represents a circle centered at the origin with a radius of 5.

Recall that parametric equations for a circle with radius $r$ can be expressed as $x = r \cos t$ and $y = r \sin t$, where $t$ is the parameter representing the angle.

Since the radius $r$ of the circle in the given equation is 5, substitute $r = 5$ into the parametric equations: $x = 5 \cos t$ and $y = 5 \sin t$.

Verify that these parametric equations satisfy the original rectangular equation by substituting $x = 5 \cos t$ and $y = 5 \sin t$ back into $x^2 + y^2 = 25$.

Simplify the expression $(5 \cos t)^2 + (5 \sin t)^2 = 25$ to confirm that it holds true, thus ensuring the parametric equations are correct.