Cogito
AP Precalculus · Chapter 7 · Lesson 1
Polar Coordinates
Locating a point by distance and direction.
12 problems · about 21 minutes · AP Precalculus 3.13, 3.14
What this lesson teaches
The student plots polar coordinates and converts between polar and rectangular form.
- A polar point is (r, θ): distance and direction.
- x = r cos θ and y = r sin θ convert to rectangular form.
- Every point has infinitely many polar names.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThe rectangular point (8, 6). What is r?
Answer 10
Why 10.
What does r represent in a polar coordinate?
Answer The distance from the origin.
Why The distance.
(5, 90°) in polar. What is the y-coordinate?
Answer 5
Why 5 sin 90°.
(5, 90°) in polar. What is the x-coordinate?
Answer 0
Why 5 cos 90°.
The rectangular point (6, 8). What is r?
Answer 10
Why √100.
Build It Up
The same ideas with more to keep track of.
3 problems(2, 180°) in polar. What is the x-coordinate?
Answer -2
Why 2 cos 180°.
The rectangular point (0, 7). What is θ in degrees?
Answer 90
Why Straight up.
(4, 270°) in polar. What is the y-coordinate?
Answer -4
Why 4 sin 270°.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich polar pairs name the same point as (3, 30°)?
Answer (3, 390 degrees); (negative 3, 210 degrees)
Why A full turn changes nothing, and a negative r reverses direction.
The rectangular point (−5, 0). What is r?
Answer 5
Why Distance is never negative.
The Distance: The rectangular point (3, 4). What is r?
Answer 5
Why 5.
The Coordinate: The polar point (6, 0°). What is its x-coordinate?
Answer 6
Why 6.