Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Find radial distance.
- Choose an angle using its quadrant.
- Check both recovered Cartesian coordinates.
Before you start
Coordinate distance, unit-circle values, and radian angles.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A point has Cartesian coordinates (−3, 3√3). Find polar coordinates with r ≥ 0 and 0 ≤ θ < 2π, then convert back to check.
Why this math matters
Cartesian coordinates describe separate horizontal and vertical moves. Polar coordinates describe distance from one origin and a direction, which is useful for rotational diagrams and radial patterns.
Set up the model
A useful answer starts with clear assumptions:
- The pole is the Cartesian origin.
- The polar axis is the positive x-axis and positive angles turn counterclockwise.
- The requested representation has nonnegative radius and angle in [0, 2π).
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Describe a point by distance and direction
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A point has Cartesian coordinates (−3, 3√3). Find polar coordinates with r ≥ 0 and 0 ≤ θ < 2π, then convert back to check.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Calculate radial distance
r = √((−3)² + (3√3)²) = √36 = 6
Radius measures straight distance from the origin. It is nonnegative under the requested convention.
Choose the correct angle
cos θ = −3/6 = −1/2; sin θ = 3√3/6 = √3/2; θ = 2π/3
Negative x and positive y require quadrant II. The 60° reference angle therefore gives 120°, not −60°.
Convert back
x = 6cos(2π/3) = −3; y = 6sin(2π/3) = 3√3
Recovering both signs and magnitudes checks the selected quadrant, not merely the ratio y/x.
The result
The requested polar coordinates are (6, 2π/3).
Adding any full turn to the angle gives the same point. A negative radius is also possible under broader conventions, but was excluded from the requested representation.
Common mistakes to catch
- arctan(y/x) alone returns −π/3 here and misses quadrant II.
- A polar pair is radius then angle; the entries do not represent x and y.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Convert polar coordinates (4, π/6) to Cartesian coordinates.
Show a hint
Use x = rcos θ and y = rsin θ.
Reveal answer and explanation
(2√3, 2)
Multiply (√3/2, 1/2) by 4.
Practice 2
Give the nonnegative-radius polar coordinates of (0, −5) with 0 ≤ θ < 2π.
Show a hint
The point lies on the negative y-axis.
Reveal answer and explanation
(5, 3π/2)
Its distance is 5 and the standard-position direction is 270°.
Take the idea with you
Use the signs of both Cartesian coordinates whenever converting a direction from a ratio.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Trace an ellipse and compare its changing speed
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