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Converting Between Polar and Rectangular Form

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 and rectangular coordinates describe the same point with different anchors. 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

Why it matters: Parametric, polar, vector, and limit ideas prepare students for motion, curves, and the rate-of-change thinking used in Calculus.

Worked example

Problem. Convert the rectangular point (3, 4) to polar radius r.

  1. Worked Example: First identify exactly what the question is asking: Convert the rectangular point (3, 4) to polar radius r.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Use r = sqrt(x^2 + y^2).
  4. r = sqrt(3^2 + 4^2) = sqrt(25).
  5. So r = 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

Practice problems

1. Convert the rectangular point (3, 4) to polar radius r.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Convert the rectangular point (3, 4) to polar radius r.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Use r = sqrt(x^2 + y^2).
  4. r = sqrt(3^2 + 4^2) = sqrt(25).
  5. So r = 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

2. Convert the rectangular point (3, 4) to polar radius r. (variation 2)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Convert the rectangular point (3, 4) to polar radius r.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Use r = sqrt(x^2 + y^2).
  4. r = sqrt(3^2 + 4^2) = sqrt(25).
  5. So r = 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

3. Convert the rectangular point (3, 4) to polar radius r. (variation 3)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Convert the rectangular point (3, 4) to polar radius r.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Use r = sqrt(x^2 + y^2).
  4. r = sqrt(3^2 + 4^2) = sqrt(25).
  5. So r = 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

4. The rectangular relationships for polar conversion include:

Choices: x = r cos(theta) and y = r sin(theta) · x = r + theta only · y = r/theta only · r = x - y

Show solution
  1. Cosine gives the horizontal component.
  2. Sine gives the vertical component.
  3. These relationships connect polar and rectangular forms.

Answer: x = r cos(theta) and y = r sin(theta)

5. The rectangular relationships for polar conversion include: (variation 2)

Choices: x = r cos(theta) and y = r sin(theta) · x = r + theta only · y = r/theta only · r = x - y

Show solution
  1. Cosine gives the horizontal component.
  2. Sine gives the vertical component.
  3. These relationships connect polar and rectangular forms.

Answer: x = r cos(theta) and y = r sin(theta)

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