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Pre-Calculus Parametric and polar

Converting between parametric and rectangular

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Theory

To turn parametric equations into a single rectangular equation in \(x\) and \(y\), eliminate the parameter \(t\):

  1. Solve one equation for \(t\).
  2. Substitute into the other.
  3. Simplify to a relation in \(x\) and \(y\).

When \(t\) appears through trig functions, use an identity such as \(\cos^2 t+\sin^2 t=1\) instead.

Carry over restrictions. If \(t\) is limited (say \(t\ge 0\)), the rectangular curve inherits a matching restriction on \(x\) or \(y\).
Eliminate the parameter Eliminate the parameter Eliminate the parameter 1. solve one equation for t 2. substitute into the other result: a relation in x, y
The elimination process.
Parametric to rectangular Eliminating t from x equals t and y equals t squared gives the rectangular equation y equals x squared. x y y = x²
\(x=t,\ y=t^2\) becomes \(y=x^2\).

Two elimination routes:

\[\text{solve for } t \text{ and substitute};\qquad \cos^2 t+\sin^2 t=1 \text{ for trig forms}\]
eliminate t by substitution, or use the Pythagorean identity for trigonometric parametrizations
The rectangular equation loses the direction and timing that the parametric form carried.

How to convert to rectangular form

  1. Pick the easier equation and solve it for \(t\).
  2. Substitute into the other equation.
  3. Use an identity if \(t\) is inside trig functions.
  4. State any restriction inherited from the \(t\)-range.
Example 1 — Eliminate t
Convert \(x=t,\ y=t^2\) to rectangular form.
Solution

Here \(t=x\); substitute into \(y=t^2\).

\(y\)\(=\)\(t^2=x^2\)
rectangular form is y equals x squared
Example 2 — Solve then substitute
Convert \(x=2t+1,\ y=t-3\).
Solution

Solve the simpler equation for \(t\): \(t=y+3\); substitute.

\(x\)\(=\)\(2(y+3)+1\)
\(=\)\(2y+7\)

So \(x=2y+7\).

rectangular form is x equals 2 y plus 7
Example 3 — Use a trig identity
Convert \(x=3\cos t,\ y=3\sin t\).
Solution

Isolate \(\cos t,\sin t\) and use \(\cos^2+\sin^2=1\).

\(\left(\dfrac{x}{3}\right)^2+\left(\dfrac{y}{3}\right)^2\)\(=\)\(1\)
\(x^2+y^2\)\(=\)\(9\)

A circle of radius 3.

rectangular form is x squared plus y squared equals 9
Example 4 — Watch the restricted range
For \(x=t^2,\ y=t\) with \(t\ge 0\), what is the rectangular equation and restriction?
Solution

\(t=y\), so \(x=y^2\); but \(t\ge 0\) means \(y\ge 0\).

\(x\)\(=\)\(y^2,\quad y\ge 0\)

Only the upper half of the sideways parabola.

x equals y squared with y at least 0

Common pitfalls

Keep the restriction. A limited \(t\) restricts the final curve; don't report the whole graph.
Use identities for trig forms. You usually can't solve \(x=\cos t\) for \(t\) cleanly.
Rectangular form drops direction. The orientation of travel is lost in conversion.

Frequently asked questions

How do you convert parametric equations to rectangular form?

Eliminate the parameter: solve one equation for \(t\) and substitute into the other, or use a trig identity.

How do you eliminate t when trig functions are involved?

Isolate \(\cos t\) and \(\sin t\), then apply \(\cos^2 t+\sin^2 t=1\).

Do you keep the parameter's restriction?

Yes. A restricted \(t\)-interval limits the rectangular curve to the corresponding piece.

What is lost when converting to rectangular form?

The direction and speed of tracing; rectangular form shows only the shape.

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