Cogito
Precalculus · Chapter 5 · Lesson 2
Graphs of Polar Equations
Curves that are hard in x and y and easy in r and θ.
12 problems · about 21 minutes · TEKS P.3.D
What this lesson teaches
The student identifies and sketches circles, roses, cardioids and spirals from polar equations.
- r = a constant is a circle; θ = a constant is a line through the origin.
- r = a cos(nθ) is a rose: n petals for odd n, 2n for even n.
- r = a + b cos θ is a limaçon, and a cardioid when a equals b.
Warm Up
Straightforward practice. Get the method working first.
5 problemsr = 5 cos(3θ). How many petals?
Answer 3
Why 3.
What curve does r = 8 draw?
Answer A circle of radius 8.
Why A circle of radius 8.
r = 3 cos(5θ). How many petals?
Answer 5
Why n is odd.
r = 3 cos(2θ). How many petals?
Answer 4
Why n is even, so 2n.
r = 9. What is the radius?
Answer 9
Why A constant r is a circle.
Build It Up
The same ideas with more to keep track of.
3 problemsr = 5 cos(7θ). How many petals?
Answer 7
Why Odd n.
r = 4 cos(6θ). How many petals?
Answer 12
Why Even n, so 2n.
r = 3 + 3 cos θ. Is it a cardioid? Enter 1 for yes, 0 for no.
Answer 1
Why a equals b.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each polar equation with its curve.
Answer r = 6 → A circle; θ = 45° → A line through the origin; r = θ → A spiral
Why A fixed distance, a fixed direction, and a distance that grows with the angle.
r = 4 cos(3θ). What is r when θ = 0°?
Answer 4
Why cos 0° = 1.
The Circle: r = 7. What is the radius of this circle?
Answer 7
Why 7.
The Petals: r = 2 cos(4θ). How many petals does this rose have?
Answer 8
Why 8.