Polar coordinates locate a point using a radius \(r\) and an angle \(\theta\) rather than Cartesian coordinates \((x,y)\). The conversion formulas are
\[x=r\cos\theta,\qquad y=r\sin\theta.\]
A polar graph is usually written as \(r=f(\theta)\). In OnlineGraphing, the angular parameter may be entered as \(t\), so a formula such as \(r=3\cos(4\theta)\) can be entered as 3*cos(4*t) in polar mode.
1. Graph a basic circle
The equation \(r=2\) gives a circle of radius 2 centered at the origin. As the angle runs from \(0\) to \(2\pi\), the radius remains constant.
The equation \(r=2\cos\theta\) gives a different circle:
\[r^2=2r\cos\theta\Longrightarrow x^2+y^2=2x\Longrightarrow (x-1)^2+y^2=1.\]
It is centered at \((1,0)\) with radius 1.
2. Predict the number of petals in a rose curve
Rose curves have the form
\[r=a\cos(n\theta)\quad\text{or}\quad r=a\sin(n\theta).\]
- If \(n\) is odd, the curve has \(n\) petals.
- If \(n\) is even, the curve has \(2n\) petals.
- The absolute value \(|a|\) is the maximum petal length.
Thus \(r=3\cos(4\theta)\) has eight petals, each reaching radius 3. Enter 3*cos(4*t) and use an angle interval such as \(0\le t\le2\pi\).
3. Understand negative radius
When \(r<0\), the point is plotted in the direction opposite the angle. Specifically, \((r,\theta)\) represents the same point as \((-r,\theta+\pi)\). Negative radius is not an error; it is what allows many polar curves to form loops and alternating petals.
4. Use symmetry before plotting
Common symmetry tests include:
- Replace \(\theta\) by \(-\theta\) to test symmetry about the polar axis.
- Replace \(\theta\) by \(\pi-\theta\) to test symmetry about the vertical axis.
- Replace \(\theta\) by \(\theta+\pi\) to test symmetry about the origin.
These tests help predict the finished curve and reveal whether a smaller angle interval can trace it completely.
5. Example: an inner-loop limaçon
Try \(r=1+2\cos\theta\), entered as 1+2*cos(t). Because the cosine coefficient is larger than the constant term, the radius becomes negative for part of the cycle, creating an inner loop.
6. Choose an appropriate viewing window
For a curve with maximum radius \(R\), a Cartesian viewing window slightly larger than \([-R,R]\times[-R,R]\) is usually appropriate. Keep equal axis scale, or petals that should be identical may appear stretched.
Common mistakes
- Using the standard function mode instead of polar mode.
- Mixing degrees and radians.
- Expecting an even value of \(n\) to produce only \(n\) petals.
- Treating negative radius as invalid.
- Using an angle interval too short to complete the curve.
Experiment in the Graphing Calculator. To compare polar and Cartesian descriptions, read circles, ellipses, and implicit equations.