Polar Coordinates
A free Precalculus lesson from the “Parametric, Polar, Vectors, and Intro to Limits” unit, with a worked example and practice problems including step-by-step solutions.
Polar coordinates describe a point by distance from the origin and direction from the positive x-axis. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Locate points using radius and angle
- Use polar coordinates in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. In the polar point (3, pi/2), what does 3 represent?
- Worked Example: First identify exactly what the question is asking: In the polar point (3, pi/2), what does 3 represent?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The first polar coordinate is r.
- r measures distance from the origin.
- The angle gives direction.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: distance from the origin
Practice problems
1. In the polar point (3, pi/2), what does 3 represent?
Choices: distance from the origin · x-coordinate · slope · area
Show solution
- Warm-up: First identify exactly what the question is asking: In the polar point (3, pi/2), what does 3 represent?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The first polar coordinate is r.
- r measures distance from the origin.
- The angle gives direction.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: distance from the origin
2. In the polar point (4, pi/2), what does 4 represent?
Choices: distance from the origin · x-coordinate · slope · area
Show solution
- Warm-up: First identify exactly what the question is asking: In the polar point (4, pi/2), what does 4 represent?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The first polar coordinate is r.
- r measures distance from the origin.
- The angle gives direction.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: distance from the origin
3. In the polar point (5, pi/2), what does 5 represent?
Choices: distance from the origin · x-coordinate · slope · area
Show solution
- Core Practice: First identify exactly what the question is asking: In the polar point (5, pi/2), what does 5 represent?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The first polar coordinate is r.
- r measures distance from the origin.
- The angle gives direction.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: distance from the origin
4. A negative r-value in polar coordinates usually means:
Choices: move in the opposite direction from the angle · the point is impossible · the angle must be zero · the radius is a y-intercept
Show solution
- Polar coordinates can use negative r.
- Negative radius reverses direction through the origin.
- This is one source of quadrant mistakes.
Answer: move in the opposite direction from the angle
5. A negative r-value in polar coordinates usually means: (variation 2)
Choices: move in the opposite direction from the angle · the point is impossible · the angle must be zero · the radius is a y-intercept
Show solution
- Polar coordinates can use negative r.
- Negative radius reverses direction through the origin.
- This is one source of quadrant mistakes.
Answer: move in the opposite direction from the angle
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