Why Make a Key-Angle Table?
A polar equation gives a radius \(r\) for each angle \(\theta\). The equation alone may not make the curve’s shape easy to see. A table of carefully chosen angles and radii turns the equation into a set of points that can be plotted and connected in order.
The important word is carefully. A table made only from equally spaced angles can miss where the radius is zero, where it changes sign, or where it reaches a maximum or minimum. Angles connected to those features are usually more useful than a large number of arbitrary samples.
For a polar point \((r,\theta)\), its Cartesian location is \(x=r\cos\theta\) and \(y=r\sin\theta\). These coordinates are useful for placing points on a sketch, but a polar graph is traced by varying \(\theta\) and following the points in that order.
Choosing Angles That Reveal the Curve
Begin by deciding on an interval of angles that traces the portion of the curve you want to sketch. For a full turn, a common interval is \(0\leq\theta\leq2\pi\). Then choose angles that help reveal the curve’s behavior, rather than relying only on equal spacing.
- Include convenient reference angles, such as \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).
- Find angles where \(r=0\). The curve is at the pole, the origin, at those angles.
- Look for intervals where \(r\) is positive or negative. A sign change can affect how plotted points lie around the pole.
- Include angles where \(r\) reaches important high or low values. Solving \(r'(\theta)=0\) can help locate these, when the derivative is available and the angle is in the interval.
- If a symmetry test establishes symmetry, use it to reduce the work, but still plot points in a way that makes the angle’s role clear.
After choosing angles, evaluate the radius exactly when possible. A decimal approximation may be useful for plotting, but exact values make it easier to detect zeros and sign changes without rounding error.
What a Negative Radius Means
A negative radius does not mean that a point should be plotted at a negative distance while keeping the same direction. Instead, it places the point in the direction opposite to the angle.
To see why, use the Cartesian coordinates of the polar point. The point \((-a,\theta)\) has coordinates \((-a\cos\theta,-a\sin\theta)\). Since \(\cos(\theta+\pi)=-\cos\theta\) and \(\sin(\theta+\pi)=-\sin\theta\), the point \((a,\theta+\pi)\) has coordinates \((-a\cos\theta,-a\sin\theta)\) as well. The two polar descriptions locate the same point.
Worked Example: A Curve with Positive Radius
Worked Example: Plotting Reference Angles
Make a key-angle table for \(r=2+\cos\theta\) on \(0\leq\theta\leq2\pi\), then describe how to use it in a sketch.
Use the four standard quarter-turn angles and include the endpoint \(2\pi\). Evaluate \(r\) at each angle:
| \(\theta\) | \(r\) | Point or direction from the pole |
|---|---|---|
| \(0\) | \(3\) | 3 units to the right |
| \(\frac{\pi}{2}\) | \(2\) | 2 units upward |
| \(\pi\) | \(1\) | 1 unit to the left |
| \(\frac{3\pi}{2}\) | \(2\) | 2 units downward |
| \(2\pi\) | \(3\) | 3 units to the right |
All the radii are positive, so each point lies in the direction named by its angle. Plot the points in angle order and sketch a smooth curve through them, returning to the starting point at \(2\pi\). The table gives useful landmarks, not a guarantee that the curve is exactly determined between every pair of sampled angles.
Check: Substituting \(\theta=0\) and \(\theta=2\pi\) gives \(r=3\) at both endpoints, as expected for the same direction after one full turn. The listed values also follow directly from the cosine values at the four reference angles.
Worked Example: Marking a Pole Crossing and Negative Radius
Worked Example: Tracking the Sign of the Radius
Make a key-angle table for \(r=1+2\cos\theta\) on \(0\leq\theta\leq2\pi\). Identify the pole angles and plot the negative-radius entry correctly.
First find where the radius is zero:
On \(0\leq\theta\leq2\pi\), this occurs at \(\theta=\frac{2\pi}{3}\) and \(\theta=\frac{4\pi}{3}\). Choose these angles along with convenient angles on either side, plus \(\pi\) and the endpoints.
| \(\theta\) | \(r\) | Plotting instruction |
|---|---|---|
| \(0\) | \(3\) | 3 units along angle \(0\) |
| \(\frac{\pi}{3}\) | \(2\) | 2 units along angle \(\frac{\pi}{3}\) |
| \(\frac{2\pi}{3}\) | \(0\) | At the pole |
| \(\pi\) | \(-1\) | 1 unit along angle \(2\pi\) |
| \(\frac{4\pi}{3}\) | \(0\) | At the pole |
| \(\frac{5\pi}{3}\) | \(2\) | 2 units along angle \(\frac{5\pi}{3}\) |
| \(2\pi\) | \(3\) | 3 units along angle \(2\pi\) |
At \(\theta=\pi\), the radius is \(-1\). The plotted point is therefore \((1,0)\) in Cartesian coordinates: one unit in the direction \(\pi+\pi=2\pi\), not one unit to the left. The radius is positive before \(\frac{2\pi}{3}\), negative between the two zero angles, and positive after \(\frac{4\pi}{3}\). The curve reaches the pole at both zero angles.
Check: At the two proposed pole angles, \(\cos\theta=-\frac12\), so \(r=1+2(-\frac12)=0\). At \(\theta=\pi\), \(r=-1\), and the Cartesian coordinates are \(x=(-1)\cos\pi=1\), \(y=(-1)\sin\pi=0\), confirming the plotted location.
Zero Radius Does Not Always Mean a Sign Change
A zero of \(r\) always places the curve at the pole. But a zero does not necessarily separate positive radii from negative radii. To determine whether the radius changes sign, check its values on both sides of the zero, or analyze its sign over the intervals between zeros.
Worked Example: A Radius That Reaches Zero Without Becoming Negative
Make a key-angle table for \(r=1+\sin(2\theta)\) on \(0\leq\theta\leq2\pi\). Locate its zero-radius angles and decide whether the radius changes sign there.
The sine function ranges from \(-1\) to \(1\), so \(1+\sin(2\theta)\) is at least zero. It equals zero when \(\sin(2\theta)=-1\), which on this interval occurs at \(\theta=\frac{3\pi}{4}\) and \(\theta=\frac{7\pi}{4}\). Evaluate at these angles and at nearby quarter- and eighth-turn landmarks:
The radius is never negative, and it is positive on either side of each zero. Thus the curve reaches the pole at \(\frac{3\pi}{4}\) and \(\frac{7\pi}{4}\), but the radius does not change sign at either angle. Plot the zero entries at the pole and use the neighboring entries to show how the curve approaches and leaves it.
Check: The minimum possible value of \(1+\sin(2\theta)\) is \(1-1=0\), confirming that no table entry can represent a negative radius. At each stated zero, the sine term is \(-1\), so the radius is exactly zero.
A Reliable Workflow for Sketching
A key-angle table is most useful when it is part of a consistent process. The following steps keep the algebra, sign analysis, and plotting connected.
Decide which interval traces the curve or portion being sketched. Include endpoints that represent the full angular range you need.
Include reference angles, zeros of \(r\), sign-change boundaries, and useful high or low values. Add angles where symmetry can help organize the sketch.
Substitute each angle carefully. Record exact values where possible, and determine whether each radius is positive, zero, or negative.
Place positive radii along their listed directions, negative radii in the opposite directions, and zero radii at the pole. Follow the increasing angles as you connect the points.
Check that all pole entries are at the origin, the negative-radius points are reversed, and repeated endpoints or symmetry are consistent with the equation.
Common Mistake: Keeping the Direction of a Negative Radius
Suppose a table gives \(r=-2\) at \(\theta=\frac{\pi}{3}\). A tempting but incorrect calculation is to treat the point as two units along the angle \(\frac{\pi}{3}\), perhaps by replacing \(r\) with \(|r|\) while leaving the angle unchanged. That plots the wrong point.
The correct Cartesian coordinates are \(x=(-2)\cos\frac{\pi}{3}=-1\) and \(y=(-2)\sin\frac{\pi}{3}=-\sqrt{3}\). They agree with a point two units along \(\frac{4\pi}{3}\), since \(2\cos\frac{4\pi}{3}=-1\) and \(2\sin\frac{4\pi}{3}=-\sqrt{3}\).
Correction: Keep the negative sign in the radius-to-coordinate calculation, or replace a negative radius by its positive magnitude and add \(\pi\) to the angle. Do not change the sign of \(r\) without changing the direction.
Reading a Table with Appropriate Caution
A table records selected points, not every point on a continuous curve. Connecting points with straight segments can give a poor sketch, especially if the chosen angles are too far apart or omit a zero. Use the equation’s behavior between entries to guide a smooth connection. When the sign of \(r\) may change, identify the zeros first; a sketch that ignores them can place an entire portion on the wrong side of the pole.
Similarly, a point at the pole alone does not tell you whether the curve crosses through the pole or only reaches it and returns. Compare the signs of \(r\) on either side of the zero. If the radius changes sign, it is on opposite sides of zero as the angle passes through that value. If it stays nonnegative or nonpositive, the sign does not switch there. The table and sign analysis together provide information that a plotted point alone cannot.
Check Your Understanding
Use the definitions and procedures in this tutorial to answer the following questions.
- For \(r=2-\sin\theta\), what are the radii at \(\theta=0\) and \(\theta=\frac{\pi}{2}\)?
- Where should the point \((-3,\frac{\pi}{6})\) be plotted using a positive radius and an equivalent angle?
- What equation do you solve to identify pole angles for a polar curve \(r=f(\theta)\)?
- If the radius is positive on both sides of a zero, has it changed sign at that zero?
- Why is it useful to plot table entries in increasing angle order rather than as an unordered collection of points?
