A polar point is written as (r, theta), where r tells how far to move and theta tells which direction to face. If r is negative, the point is plotted in the opposite direction from the angle. Polar curves such as roses, cardioids, and limacons are traced by letting theta vary and calculating r.
Calculus tools such as derivatives and integrals can find tangent slopes, enclosed areas, and arc lengths for these curves.
Key Facts
- Polar to rectangular: x = r cos(theta), y = r sin(theta)
- Rectangular to polar: r^2 = x^2 + y^2, tan(theta) = y/x
- Polar area formula: A = (1/2) integral from alpha to beta of r^2 dtheta
- Slope of a polar curve: dy/dx = (r' sin(theta) + r cos(theta)) / (r' cos(theta) - r sin(theta))
- Rose curve forms: r = a cos(n theta) or r = a sin(n theta)
- Cardioid forms: r = a(1 + cos(theta)), r = a(1 - cos(theta)), r = a(1 + sin(theta)), or r = a(1 - sin(theta))
Vocabulary
- Polar coordinate
- A coordinate written as (r, theta), where r is distance from the pole and theta is the angle from the polar axis.
- Pole
- The origin in the polar coordinate system, where r = 0.
- Polar axis
- The reference ray in polar coordinates, usually the positive x-axis.
- Rose curve
- A polar curve shaped like petals, usually given by r = a cos(n theta) or r = a sin(n theta).
- Cardioid
- A heart-shaped polar curve with a cusp at the pole, commonly written as r = a(1 ± cos(theta)) or r = a(1 ± sin(theta)).
Common Mistakes to Avoid
- Treating theta as a y-coordinate is wrong because theta is an angle, not a vertical distance.
- Forgetting that negative r reverses direction is wrong because (r, theta) and (-r, theta + pi) describe the same point.
- Using A = integral r dtheta for polar area is wrong because the area of each small sector depends on r^2, so the correct formula is A = (1/2) integral r^2 dtheta.
- Assuming a rose curve always has n petals is wrong because r = a cos(n theta) or r = a sin(n theta) has n petals when n is odd and 2n petals when n is even.
Practice Questions
- 1 Convert the polar point (6, pi/3) to rectangular coordinates.
- 2 Find the area enclosed by one petal of the rose curve r = 4 sin(3 theta).
- 3 Explain how you can predict the symmetry of the curve r = 2 + 2 cos(theta) without plotting many points.