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Pre-Calculus Parametric and polar

Converting between polar and rectangular

20 practice questions 0 video lessons Theory + worked examples

Converting Between Polar and Rectangular

California Pre-Calculus • Standard G-GPE • Parametric & Polar

Converting Between Polar and Rectangular is a topic in Parametric & Polar in the California Common Core State Standards. It is aligned to Standard G-GPE, which requires students to convert between polar and rectangular coordinates.

Converting between polar and rectangular uses \(x=r\cos\theta\), \(y=r\sin\theta\), \(r^2=x^2+y^2\), and \(\tan\theta=\dfrac{y}{x}\).

California Pre-Calculus › Parametric & Polar › Converting Between Polar and Rectangular  —  Standard G-GPE

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Theory

The same point has both rectangular coordinates \((x,y)\) and polar coordinates \((r,\theta)\), linked by a right triangle:

\[x=r\cos\theta,\qquad y=r\sin\theta,\]
\[r^2=x^2+y^2,\qquad \tan\theta=\dfrac{y}{x}.\]

Use the first pair to go polar \(\to\) rectangular, and the second to go rectangular \(\to\) polar (minding the quadrant for \(\theta\)).

Converting an equation often means multiplying by \(r\) or substituting so that \(r\cos\theta\) becomes \(x\), etc.
Polar and rectangular coordinates The same point has rectangular coordinates x, y and polar coordinates r, theta, linked by a right triangle. (x,y)=(r,θ) x y r
The same point as \((x,y)\) and \((r,\theta)\), joined by a right triangle.
Conversion Conversion Conversion x = r cos θ, y = r sin θ r² = x² + y² tan θ = yx
The conversion formulas.

The conversion relationships:

\[x=r\cos\theta,\quad y=r\sin\theta,\quad r^2=x^2+y^2,\quad \tan\theta=\dfrac{y}{x}\]
x equals r cosine theta, y equals r sine theta, r squared equals x squared plus y squared, tangent theta equals y over x
For \(\theta\), check the quadrant — \(\arctan\) alone can be off by \(180^\circ\).

How to convert coordinates

  1. Polar \(\to\) rectangular: \(x=r\cos\theta,\ y=r\sin\theta\).
  2. Rectangular \(\to\) polar: \(r=\sqrt{x^2+y^2}\), \(\theta=\arctan\dfrac{y}{x}\) (fix the quadrant).
  3. Equations: multiply by \(r\) or substitute to swap \(r\cos\theta\leftrightarrow x\), \(r^2\leftrightarrow x^2+y^2\).
Example 1 — Polar to rectangular
Convert \((4,\ 60^\circ)\) to rectangular form.
Solution

Use \(x=r\cos\theta,\ y=r\sin\theta\).

\(x\)\(=\)\(4\cos 60^\circ=2\)
\(y\)\(=\)\(4\sin 60^\circ=2\sqrt3\)
rectangular form is 2 comma 2 root 3
Example 2 — Rectangular to polar
Convert \((1,1)\) to polar form.
Solution

Find \(r\) and \(\theta\).

\(r\)\(=\)\(\sqrt{1^2+1^2}=\sqrt2\)
\(\theta\)\(=\)\(\arctan\dfrac{1}{1}=45^\circ\)

So \((\sqrt2,\ 45^\circ)\).

polar form is root 2 comma 45 degrees
Example 3 — Convert an equation
Write \(r=2\cos\theta\) in rectangular form.
Solution

Multiply by \(r\) and substitute \(r^2=x^2+y^2\), \(r\cos\theta=x\).

\(r^2\)\(=\)\(2r\cos\theta\)
\(x^2+y^2\)\(=\)\(2x\)

A circle: \((x-1)^2+y^2=1\).

rectangular form is a circle x minus 1 squared plus y squared equals 1
Example 4 — A vertical line
Convert \(x=3\) to polar form.
Solution

Substitute \(x=r\cos\theta\).

\(r\cos\theta\)\(=\)\(3\)
\(r\)\(=\)\(\dfrac{3}{\cos\theta}\)
polar form is r equals 3 over cosine theta

Common pitfalls

Fix the quadrant for \(\theta\). \(\arctan\dfrac{y}{x}\) may need \(\pm 180^\circ\).
Multiply by \(r\) to convert equations so that \(r\cos\theta\) becomes \(x\).
\(r^2=x^2+y^2\), not \(x+y\). Keep the squares.

Frequently asked questions

How do you convert polar to rectangular coordinates?

Use \(x=r\cos\theta\) and \(y=r\sin\theta\).

How do you convert rectangular to polar coordinates?

\(r=\sqrt{x^2+y^2}\) and \(\theta=\arctan\dfrac{y}{x}\), adjusting for the quadrant.

How do you convert a polar equation to rectangular form?

Multiply through by \(r\) or substitute so that \(r\cos\theta\to x\), \(r\sin\theta\to y\), and \(r^2\to x^2+y^2\).

Why check the quadrant when finding theta?

Because \(\arctan\) only returns angles in two quadrants; the actual point may require adding \(180^\circ\).

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