Convert Polar Equations to Rectangular Form

Step-by-step examples: circle, line, and ray — with formulas and practice

Overview

Equations written in polar form can be converted into rectangular (Cartesian) form using the standard relationships between polar and rectangular coordinates. Polar coordinates of a point are denoted \( (R, t) \), where \( R \) is the radial distance and \( t \) is the angular coordinate.

Conversion Formulas

The core relationships between rectangular coordinates \( (x, y) \) and polar coordinates \( (R, t) \):

\[ x = R\cos t, \qquad y = R\sin t, \qquad R^2 = x^2 + y^2, \qquad \tan t = \frac{y}{x} \]

Practical tip: multiplying the polar equation by \( R \) is often the key first step — it produces \( R^2 \) (which becomes \( x^2 + y^2 \)) or \( R\sin t \) / \( R\cos t \) (which become \( y \) / \( x \)). Also note that \( \tan t = y/x \) only fixes \( t \) up to \( \pi \), so quadrant and ray restrictions must be handled separately (see Problem 3).

Problems and Solutions

Problem 1 — Convert \(R = 4\sin t\) to rectangular form

Convert the polar equation \( R = 4\sin t \) to rectangular form and identify the curve.

Solution to Problem 1

Step 1: Multiply both sides by \( R \):

\[ R^2 = 4R\sin t \]

Step 2: Substitute \( R^2 = x^2 + y^2 \) and \( y = R\sin t \):

\[ x^2 + y^2 = 4y \]

Step 3: Rearrange all terms to one side:

\[ x^2 + y^2 - 4y = 0 \]

Step 4: Complete the square in \( y \):

\[ x^2 + (y^2 - 4y + 4) = 4 \] \[ x^2 + (y - 2)^2 = 4 \]

Conclusion: a circle centered at \( (0, 2) \) with radius \( 2 \).

Problem 2 — Convert \(R(-2\sin t + 3\cos t) = 2\) to rectangular form

Convert the polar equation \( R(-2\sin t + 3\cos t) = 2 \) to rectangular form and identify the curve.

Solution to Problem 2

Step 1: Expand the left-hand side:

\[ -2R\sin t + 3R\cos t = 2 \]

Step 2: Substitute \( y = R\sin t \) and \( x = R\cos t \):

\[ -2y + 3x = 2 \]

Step 3: Write in standard linear form:

\[ 3x - 2y = 2 \qquad \text{or} \qquad y = \tfrac{3}{2}x - 1 \]

Conclusion: a straight line with slope \( \tfrac{3}{2} \) and y-intercept \( (0, -1) \).

Problem 3 — Convert \(t + \tfrac{\pi}{4} = 0\) to rectangular form

Convert the polar equation \( t + \tfrac{\pi}{4} = 0 \) to rectangular form and identify the curve.

Solution to Problem 3

Step 1: Solve for \( t \):

\[ t = -\frac{\pi}{4} \]

Step 2: Take the tangent of both sides:

\[ \tan t = \tan\!\left(-\frac{\pi}{4}\right) = -1 \]

Step 3: Use the identity \( \tan t = \tfrac{y}{x} \):

\[ \frac{y}{x} = -1 \qquad (x \ne 0) \]

Step 4: Solve for \( y \):

\[ y = -x \qquad (x \ne 0) \]

Careful — this is a ray, not a full line. The polar equation \( t = -\tfrac{\pi}{4} \) (with \( R \ge 0 \)) represents the ray from the origin into the fourth quadrant, at angle \( -\tfrac{\pi}{4} \). In rectangular form this is the portion of the line \( y = -x \) with \( x > 0 \).

The full line \( y = -x \) corresponds to the union of the two rays \( t = -\tfrac{\pi}{4} \) and \( t = \tfrac{3\pi}{4} \) — so if negative \( R \) is allowed, the polar equation covers the entire line.

Practice Exercises

Convert each polar equation to rectangular form and identify the curve.

Exercise 1 — \(R = 6\cos t\)

Answer

Multiply by \(R\): \(R^2 = 6R\cos t \Rightarrow x^2 + y^2 = 6x \Rightarrow (x - 3)^2 + y^2 = 9\).

Circle centered at \((3, 0)\), radius \(3\).

Exercise 2 — \(R = 2\)

Answer

Square both sides: \(R^2 = 4 \Rightarrow x^2 + y^2 = 4\).

Circle centered at the origin, radius \(2\).

Exercise 3 — \(R = \dfrac{4}{\cos t + \sin t}\)

Answer

Multiply by the denominator: \(R\cos t + R\sin t = 4\).

Substitute \(x = R\cos t\) and \(y = R\sin t\): \(x + y = 4\).

Line with slope \(-1\) and y-intercept \((0, 4)\).

Frequently Asked Questions

What are the formulas for converting polar equations to rectangular form?

Use \(x = R\cos t\), \(y = R\sin t\), \(R^2 = x^2 + y^2\), and \(\tan t = y/x\). Multiplying by \(R\) is often the key first step, since it lets you replace \(R^2\) with \(x^2 + y^2\) and \(R\sin t\) or \(R\cos t\) with \(y\) or \(x\).

How do you convert \(R = 4\sin t\) to rectangular form?

Multiply both sides by \(R\): \(R^2 = 4R\sin t\). Substitute \(R^2 = x^2 + y^2\) and \(R\sin t = y\): \(x^2 + y^2 = 4y\). Complete the square to get \(x^2 + (y - 2)^2 = 4\) — a circle centered at \((0, 2)\) with radius \(2\).

How do you convert a polar angle equation like \(t = \pi/4\) to rectangular form?

Take \(\tan\) of both sides: \(\tan t = \tan(\pi/4) = 1\). Since \(\tan t = y/x\), this gives \(y/x = 1\), i.e. \(y = x\) with \(x > 0\) (the ray from the origin at \(45°\)). Without the \(x > 0\) restriction, \(y = x\) would describe the full line — which corresponds to both the ray at \(t = \pi/4\) and the opposite ray at \(t = 5\pi/4\).

Why does the polar-to-rectangular conversion sometimes give a larger curve?

Because polar equations can trace the same point multiple times as \(t\) varies, and can also produce points with \(R < 0\) (which flip direction). The rectangular form describes the full set of points reached by the polar equation, which is sometimes a larger curve than a single traversal suggests.

More Practice on Polar Coordinates

Last updated: October 17, 2026