Recognizing the Shape from Its Equation
A polar equation such as \(r=3+2\cos\theta\) describes a curve by giving the distance \(r\) from the pole for each angle \(\theta\). The curve’s overall shape depends on how the constant term and the trigonometric term compare. A relatively large constant gives a rounded curve; when the trigonometric term is large enough, the graph can develop a dimple or an inner loop.
For equations in the standard form \(r=a+b\cos\theta\) or \(r=a+b\sin\theta\), sketching begins with three decisions: identify the symmetry axis, calculate a few key radii, and compare the positive coefficients \(a\) and \(b\). The ratio \(a/b\) is especially useful when \(a>0\) and \(b>0\).
Unless stated otherwise, the classifications below assume \(a>0\) and \(b>0\). A negative radius is plotted in the direction opposite the angle’s ray: the point \((r,\theta)\) with \(r<0\) is plotted at distance \(|r|\) along angle \(\theta+\pi\). This matters when deciding whether an inner loop occurs.
Symmetry and Orientation
The cosine form \(r=a+b\cos\theta\) is symmetric about the polar axis, which is the horizontal axis. Indeed, cosine is unchanged when \(\theta\) is replaced by \(-\theta\), so the same radius occurs at angles on opposite sides of that axis. For \(a,b>0\), the largest radius occurs at \(\theta=0\), so the main bulge points to the right.
The sine form \(r=a+b\sin\theta\) is symmetric about the vertical axis. Replacing \(\theta\) by \(\pi-\theta\) leaves sine unchanged. Its largest radius occurs at \(\theta=\pi/2\), so for positive coefficients the main bulge points upward. A sine limaçon is therefore the corresponding cosine limaçon turned so its symmetry axis is vertical.
Key Angles and Classification
For the cosine form, the angles \(0,\pi/2,\pi,3\pi/2\) expose the farthest and nearest radii and the symmetry of the curve. In particular, \(r(0)=a+b\) and \(r(\pi)=a-b\). The minimum radius is therefore controlled by the difference between the coefficients. For the sine form, the corresponding useful angles are \(\pi/2,0,3\pi/2,\pi\).
| Coefficient comparison, \(a,b>0\) | Shape | What the comparison tells you |
|---|---|---|
| \(0<a<b\) | Inner-loop limaçon | \(r(\pi)=a-b<0\), so some plotted points use negative radius. |
| \(a=b\) | Cardioid | The radius reaches zero at the side opposite the main bulge. |
| \(b<a<2b\) | Dimpled limaçon | The radius stays positive, but the curve has an inward indentation. |
| \(a\geq 2b\) | Convex limaçon | The curve has no inner loop or inward dimple. |
The ratio is \(a/b\), not \(b/a\), when using this table. A value below \(1\) gives an inner loop; a value of \(1\) gives a cardioid; values between \(1\) and \(2\) give a dimpled curve; and values at least \(2\) give a convex curve.
At \(a=b\), the curve reaches the pole when the trigonometric term is at its minimum: \(\theta=\pi\) for cosine and \(\theta=3\pi/2\) for sine. This point forms the cardioid’s cusp. If \(a<b\), the minimum radius is negative instead, and the curve has an inner loop. The detailed tracing of pole crossings is a separate skill; for classification here, the sign of the minimum radius is the key warning.
Why the Dimple-to-Convex Boundary Is \(a/b=2\)
The threshold between a dimpled and convex limaçon can be justified using derivatives. Write the curve in Cartesian parametric form as \(x=r\cos\theta\), \(y=r\sin\theta\). For a polar curve, differentiating these coordinates gives a signed turning numerator \(x'y''-y'x''\). When \(r>0\) and the curve is regular, its sign indicates whether the curve bends consistently outward or bends inward somewhere.
From \(x=r\cos\theta\) and \(y=r\sin\theta\), we obtain \(x'=r'\cos\theta-r\sin\theta\) and \(y'=r'\sin\theta+r\cos\theta\).
Differentiating once more and simplifying \(x'y''-y'x''\) gives \(r^2+2(r')^2-r r''\). This formula follows by substituting the derivatives of \(x\) and \(y\); the terms involving \(\sin\theta\cos\theta\) cancel.
For \(r=a+b\cos\theta\), \(r'=-b\sin\theta\) and \(r''=-b\cos\theta\). Thus the numerator is \(a^2+2b^2+3ab\cos\theta\).
Since \(-1\leq\cos\theta\leq1\), the minimum is \(a^2+2b^2-3ab=(a-b)(a-2b)\).
For \(a>b\), the radius is positive everywhere, so the curve is traced without negative-radius portions. If \(b<a<2b\), then \(a-b>0\) and \(a-2b<0\), making the minimum turning numerator negative. The curve bends inward somewhere, producing a dimple. If \(a\geq2b\), both factors are nonnegative, so the numerator is nonnegative throughout; the curve is convex. At \(a=2b\), it reaches zero at one angle but does not become negative. This establishes the classification boundary, rather than relying only on the appearance of a sketch.
Worked Examples
Worked Example: A Cardioid in Cosine Form
Classify and sketch \(r=2+2\cos\theta\).
Here \(a=2\) and \(b=2\), so \(a/b=2/2=1\). The equation is a cardioid. Because it uses cosine and both coefficients are positive, it is symmetric about the horizontal axis and bulges to the right.
Evaluate the radius at key angles:
The farthest point is 4 units to the right. The curve reaches the pole at \(\theta=\pi\), forming the cusp opposite the bulge. The equal radii at \(\pi/2\) and \(3\pi/2\) are consistent with horizontal symmetry. Check the key classification by substituting \(a=b=2\): the minimum radius is \(a-b=2-2=0\), as required for a cardioid.
Worked Example: A Dimpled Sine Limaçon
Classify and sketch \(r=3+2\sin\theta\).
Here \(a=3\) and \(b=2\), so \(a/b=3/2=1.5\). Since \(1<1.5<2\), this is a dimpled limaçon. It is symmetric about the vertical axis and bulges upward.
Use angles that identify the top, bottom, and side points:
The greatest radius is 5 upward, and the least is 1 downward. Because that minimum is positive, the curve does not have an inner loop. The ratio places it in the dimpled category, so the sketch should include a shallow indentation on the lower side rather than a smooth convex outline. Check: \(a-b=3-2=1\), matching the minimum radius found at \(3\pi/2\).
Worked Example: An Inner-Loop Limaçon
Classify and sketch \(r=1+2\cos\theta\).
Now \(a=1\) and \(b=2\), so \(a/b=1/2=0.5<1\). This is an inner-loop limaçon. Its symmetry axis is horizontal, and its largest positive radius points right.
Calculate the radii at key angles:
The value \(-1\) at \(\theta=\pi\) is not a point one unit to the left: a negative radius is plotted in the opposite direction, one unit to the right. This negative minimum signals the inner-loop form. The equal side values of 1 agree with symmetry about the horizontal axis, and \(r(0)=3\) identifies the farthest point on the main bulge. Check the classification directly: \(a-b=1-2=-1<0\), so the radius becomes negative, consistent with the inner-loop category.
A Reliable Sketching Process
For a standard limaçon with positive coefficients, use the following sequence. A careful sketch does not require a large table: symmetry and the maximum and minimum radii establish the orientation and broad shape.
Cosine gives horizontal symmetry; sine gives vertical symmetry.
For cosine, use \(r(0)=a+b\) and \(r(\pi)=a-b\). For sine, use \(r(\pi/2)=a+b\) and \(r(3\pi/2)=a-b\).
Use the classification table to decide whether the curve has an inner loop, a cusp, a dimple, or a convex outline.
Use the symmetry axis and side-angle values to keep both halves of the sketch consistent.
Common Mistake: Reversing the Ratio
A frequent error is to compare \(b/a\) with the classification cutoffs. For example, for \(r=3+2\cos\theta\), calculating \(b/a=2/3\) and calling the curve an inner-loop limaçon is incorrect. The table uses \(a/b\), which here is \(3/2=1.5\). Since \(1<1.5<2\), the curve is dimpled. Checking the minimum radius confirms this: \(a-b=3-2=1>0\), so it never has a negative radius. Always compute the coefficients in the order named by the ratio before assigning a shape.
Check Your Understanding
For each equation, identify its symmetry, orientation, and shape. Use key-angle radii to support your answer.
- Classify \(r=4+4\cos\theta\), and find its radii at \(\theta=0\) and \(\theta=\pi\).
- Classify \(r=5+2\sin\theta\). What are its maximum and minimum radii?
- For \(r=2+3\cos\theta\), explain what the negative value at \(\theta=\pi\) tells you.
- Why does \(r=6+3\cos\theta\) lie at the boundary between the dimpled and convex classifications?
- For \(r=2+2\sin\theta\), where does the curve reach the pole, and which way does its main bulge point?
