Cogito
AP Calculus BC · Chapter 2 · Lesson 1
Polar Coordinates
Locating a point by distance and angle.
11 problems · about 24 minutes · AP Calculus BC 9.7, 9.8
What this lesson teaches
The student converts between polar and Cartesian coordinates.
- A polar coordinate is a distance and an angle.
- x = r cos θ and y = r sin θ convert to Cartesian; Pythagoras converts back.
- One point has infinitely many polar names.
Warm Up
Straightforward practice. Get the method working first.
4 problemsCartesian (6, 8). What is r?
Answer 10
Why 10.
r = 3, θ = 0°. What is y?
Answer 0
Why sin 0 is 0.
r = 4, θ = 90°. What is y?
Answer 4
Why sin 90 is 1.
Cartesian (3, 4). What is r?
Answer 5
Why Pythagoras.
Build It Up
The same ideas with more to keep track of.
3 problemsCartesian (0, 7). What is r?
Answer 7
Why On the axis.
r = 6, θ = 180°. What is x?
Answer -6
Why cos 180 is −1.
Set the angle to 270°, where cosine is 0 and sine is −1.
Answer degrees = 270
Why Three quarters of the way round.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsDoes adding 360° to θ change the point? 1 for yes, 0 for no.
Answer 0
Why A full turn.
Cartesian (0, 0). What is r?
Answer 0
Why At the origin.
Straight Out: r = 5, θ = 0°. What is x?
Answer 5
Why 5.
Straight Up: r = 5, θ = 90°. What is x?
Answer 0
Why 0.