Why Convert a Cartesian Equation to Polar Form?
Cartesian equations describe points through their horizontal and vertical coordinates \(x\) and \(y\). Polar equations describe the same points using a directed distance \(r\) from the pole and an angle \(\theta\) from the positive \(x\)-axis.
Changing coordinates does not change the curve. It changes the way the curve’s points are represented. This can make an equation shorter or reveal a useful relationship between distance and direction. Circles and lines are especially common examples.
The conversion begins with the coordinate identities. The main task is to substitute them consistently, simplify, and check that any algebraic rearrangement has not removed or added points.
The Substitution Rules
To convert a Cartesian equation, replace each occurrence of \(x\) with \(r\cos\theta\), each occurrence of \(y\) with \(r\sin\theta\), and each occurrence of \(x^2+y^2\) with \(r^2\). The last substitution follows by squaring the first two identities and adding:
Here we use the trigonometric identity \(\cos^2\theta+\sin^2\theta=1\). For example, a Cartesian equation containing \(x^2+y^2\) often becomes simpler after this replacement. A linear expression in \(x\) and \(y\), on the other hand, usually becomes a factor of \(r\) multiplied by a trigonometric expression.
We allow signed \(r\), as is standard in polar equations: a negative \(r\) represents a point in the direction opposite to the angle \(\theta\). Different polar pairs can represent the same Cartesian point. In particular, changing \(\theta\) by a full turn preserves a point, and reversing the sign of \(r\) while adding \(\pi\) to the angle also preserves it. Therefore, “equivalent equations” here means equations that represent the same set of points, not necessarily equations that admit exactly the same pairs \((r,\theta)\).
A Reliable Conversion Process
Write \(x=r\cos\theta\) and \(y=r\sin\theta\) everywhere they occur. Replace \(x^2+y^2\) by \(r^2\) when that expression appears.
Factor the resulting equation if possible. Do not divide by an expression that could equal zero until you have checked what happens in that case.
An equation may remain as a relation involving both \(r\) and \(\theta\), or it may be solved for \(r\) where that is valid.
Substitute the polar identities back into the Cartesian equation, and make sure the rearrangement has not discarded a point or introduced an invalid case.
Worked Example: Converting a Circle Through the Pole
Worked Example: Converting a Circle Through the Pole
Convert the Cartesian equation \(x^2+y^2=6x\) to polar form.
First substitute \(x=r\cos\theta\) and \(x^2+y^2=r^2\):
Factor the equation:
For a nonzero point, the second factor gives \(r=6\cos\theta\). This equation also represents the pole: when \(\cos\theta=0\), it gives \(r=0\). Thus a convenient polar equation for the curve is
Check the substitution into the original equation. If \(r=6\cos\theta\), then
The two sides agree, so every point represented by the polar equation satisfies the Cartesian equation. The factorization also shows why simply dividing by \(r\) deserves care: division excludes \(r=0\). In this example the pole is still represented by \(r=6\cos\theta\), but that should be checked rather than assumed.
Why the Substitution Preserves the Curve
The coordinate identities are not approximations. Each polar pair \((r,\theta)\) determines Cartesian coordinates by \(x=r\cos\theta\) and \(y=r\sin\theta\). Conversely, every point in the plane has a polar representation, with the pole represented by \(r=0\). Thus replacing \(x\) and \(y\) by these expressions translates the original condition on a point into a condition on its polar representation.
For instance, suppose a Cartesian equation is \(F(x,y)=0\), where \(F\) is an expression in \(x\) and \(y\). Its direct polar substitution is \(F(r\cos\theta,r\sin\theta)=0\). If a point has polar coordinates \((r,\theta)\), then the original equation holds at its Cartesian coordinates exactly when this substituted equation holds. This establishes why direct substitution works.
Further simplification needs its own care. Adding or subtracting the same expression from both sides preserves an equation. Factoring also preserves it. Dividing by a factor that may be zero can lose solutions, so those zero cases must be checked separately. Squaring both sides can introduce solutions and likewise requires a check.
Worked Example: Converting a Line
Worked Example: Converting a Line
Convert the line \(2x-3y=6\) to polar form.
Substitute \(x=r\cos\theta\) and \(y=r\sin\theta\):
Factor out \(r\):
This factored equation is a polar form of the line. It can also be solved for \(r\) when the denominator is nonzero:
The denominator cannot be zero for a point on this line: if \(2\cos\theta-3\sin\theta=0\), the factored equation would say \(0=6\), which is impossible. Therefore solving for \(r\) here does not omit a point on the line.
Check by substituting the factored equation back into the Cartesian left-hand side:
The Cartesian condition is recovered, as required.
Worked Example: Converting a Circle Not Centered at the Pole
Worked Example: Converting a Circle Not Centered at the Pole
Convert the circle \((x-2)^2+(y+1)^2=9\) to polar form.
Expand the Cartesian equation first:
Combine the constant terms and move \(9\) to the left:
Now substitute \(x^2+y^2=r^2\), \(x=r\cos\theta\), and \(y=r\sin\theta\):
This is a polar equation of the circle. To check it, substitute the coordinate identities into the expanded Cartesian equation:
The right-hand side is zero exactly when the converted equation holds. The expansion and substitution therefore preserve the original condition.
Circles and Lines: Useful Forms
Some Cartesian equations convert directly into familiar compact forms. For example, if \(x^2+y^2=a x\), substitution gives \(r^2=a r\cos\theta\). A common polar form is \(r=a\cos\theta\), with the zero-radius case checked as in the worked example. Likewise, \(x^2+y^2=b y\) gives \(r^2=b r\sin\theta\), often written \(r=b\sin\theta\).
A line \(Ax+By=C\) converts directly to
This form is safe even when the trigonometric factor might be zero. If you solve for \(r\), first consider whether \(A\cos\theta+B\sin\theta=0\). When \(C\ne 0\), such an angle cannot satisfy the line equation. When \(C=0\), the equation is a line through the pole, and dividing by that factor would discard cases; keep the undivided equation or analyze those cases separately.
Common Mistake: Using \(r\) Instead of \(r^2\)
A frequent error is replacing \(x^2+y^2\) with \(r\). The correct identity is \(x^2+y^2=r^2\), because the squares of \(x\) and \(y\) produce \(r^2\) after the trigonometric identity is applied.
Worked Example: Correcting an Incorrect Substitution
Convert \(x^2+y^2=9\). An incorrect substitution would give \(r=9\), but it has replaced a sum of squares by \(r\) rather than \(r^2\). The correct conversion is
With signed \(r\), this equation gives \(r=3\) or \(r=-3\), and both signed radii represent points at distance \(3\) from the pole. If one uses only nonnegative radii, the equation becomes \(r=3\). Check the corrected form by using \(x^2+y^2=r^2=9\), which recovers the original equation.
Another mistake is to divide by \(r\) automatically. If \(r=0\) satisfies the original equation, division may remove the pole. Factor first, identify the zero case, and check whether the simplified polar equation still represents it. Similarly, keep a line in factored form if solving for \(r\) would require dividing by a trigonometric factor that can vanish.
Why These Conversions Matter
Cartesian equations are often natural when a curve is defined by horizontal and vertical coordinates. Polar equations can be more convenient when the equation depends on distance from the pole or on direction. Converting between the forms lets you choose the representation that best expresses a given relationship.
The essential discipline is not to memorize a collection of special conversions. It is to apply the coordinate identities accurately, simplify without losing cases, and check the result in the original equation. That process works for circles, lines, and other Cartesian equations alike.
Check Your Understanding
For each question, use the coordinate identities and check any division or rearrangement.
- Convert \(x^2+y^2=8y\) to a polar equation.
- Convert \(3x+2y=5\) to polar form, first keeping it in factored form.
- Convert \(x^2+y^2-2x-4y=0\) to polar form.
- Why must \(x^2+y^2\) be replaced by \(r^2\), not \(r\)?
- What should you check before dividing a substituted equation by \(r\)?
