Polar graphing uses points described by a distance r from the origin and an angle θ from the positive x-axis. This system is useful for curves with circular or rotational patterns, such as petals, loops, spirals, and waves around a center. Instead of moving left and right with x and up and down with y, you rotate by an angle and move outward or inward by a radius.
Many shapes that look complicated in rectangular coordinates have simple polar equations.
Key Facts
- A polar point is written as (r, θ), where r is distance from the pole and θ is the angle from the polar axis.
- If r is negative, plot the point |r| units in the opposite direction from angle θ.
- Convert to rectangular coordinates with x = r cos θ and y = r sin θ.
- Convert to polar distance with r^2 = x^2 + y^2 and tan θ = y/x, adjusted for quadrant.
- Rose curves often have form r = a cos(nθ) or r = a sin(nθ); if n is odd there are n petals, and if n is even there are 2n petals.
- Cardioids and limacons often have form r = a ± b cos θ or r = a ± b sin θ, with symmetry based on sine or cosine.
Vocabulary
- Polar coordinate
- A coordinate written as (r, θ) that gives a point by its distance from the origin and its direction angle.
- Pole
- The origin of a polar coordinate system, where r = 0.
- Polar axis
- The reference ray for θ = 0, usually drawn as the positive x-axis.
- Rose curve
- A polar graph with petal-like loops, commonly written as r = a cos(nθ) or r = a sin(nθ).
- Cardioid
- A heart-shaped polar curve formed by equations such as r = a + a cos θ or r = a + a sin θ.
Common Mistakes to Avoid
- Plotting negative r as a negative distance on the same ray is wrong because negative r means move in the opposite direction from the given angle.
- Using degrees when the calculator is set to radians is wrong because values of sin θ and cos θ will not match the angle table.
- Assuming r = a cos θ and r = a sin θ have the same orientation is wrong because cosine graphs are symmetric about the polar axis while sine graphs are symmetric about the vertical line θ = 90°.
- Graphing only one or two angles is wrong because polar curves can loop, repeat, or cross the origin; a useful table should include key angles over a full interval.
Practice Questions
- 1 Make a table and plot the polar equation r = 2 cos θ for θ = 0°, 30°, 60°, 90°, 120°, 180°. What basic shape does the graph form?
- 2 For r = 3 sin(2θ), find r when θ = 0°, 30°, 45°, 60°, and 90°. Use the values to sketch the first part of the rose curve.
- 3 Explain how the graph of r = 2 + 2 cos θ differs from the graph of r = 2 + 2 sin θ in orientation and symmetry.