The Coordinate Converter
Topics 10.6, 10.7 -- Rectangular and Polar Coordinate Conversions
Step 1 of 6
A friend lives at the point (3, 4) on this grid neighborhood. Describe how to get there using only streets -- how many blocks east and how many blocks north?
Step 2 of 6
Now describe the same point by pointing and giving a distance. A ray draws from the origin to your friend's house. How far is it, and at what angle from east?
What is the distance from the origin to the point (3, 4)?
Step 3 of 6
The point, the origin, and the foot of the perpendicular form a right triangle. The hypotenuse is \(r\), the angle is \(\theta\). From that triangle, what are the two conversion formulas from polar to rectangular?
Select the correct pair of formulas:
Step 4 of 6
Convert the point \((-3, -4)\) to polar. First, compute \(\tan\theta = \frac{y}{x} = \frac{-4}{-3} = \frac{4}{3}\). Your calculator gives \(\arctan\!\left(\frac{4}{3}\right) \approx 53.1^\circ\). Is that the right angle?
Step 5 of 6
Convert the equation \(x^2 + y^2 = 25\) to polar form. Substitute \(x = r\cos\theta\) and \(y = r\sin\theta\), then simplify.
What does the equation become in polar form?
Step 6 of 6
Convert the equation \(y = x\) to polar form. What does it become?
Explore: Dual Coordinate View
Click on either grid to place a point. The equivalent appears on the other grid with the conversion worked out step by step.
Rectangular (x, y)
Polar (r, θ)
Rectangular
Polar
Place a point to see the conversion steps.
Enter Coordinates Directly
Rectangular coordinates describe a point by how far east/west (x) and how far north/south (y). Polar coordinates describe the same point by distance from the origin (r) and angle from the positive x-axis (θ).
Try This
Starter
Part A: Convert the rectangular point \((-3, -4)\) to polar coordinates with \(\theta\) in \([0, 2\pi)\). Show the quadrant check.
What is \(r\)?
What is \(\theta\), after the quadrant check?
Part B: Convert the polar point \(\left(6, \frac{2\pi}{3}\right)\) to rectangular coordinates.
Stretch
Convert \(x^2 + y^2 = 6x\) to polar form and simplify as far as you can. What shape is it?
Hint: Substitute \(x = r\cos\theta\) and \(y = r\sin\theta\). Remember that \(x^2 + y^2 = r^2\) and \(x = r\cos\theta\).
What is the simplified polar equation?
What shape is it?
Challenge
Convert \(r = \dfrac{4}{2 - \cos\theta}\) to rectangular form.
Hint: Multiply both sides by \((2 - \cos\theta)\), then use \(r\cos\theta = x\) and \(r = \sqrt{x^2 + y^2}\). It will be messy.
Which rectangular form is correct?
Which form would you rather graph, and why?
- Convert points between rectangular \((x, y)\) and polar \((r, \theta)\) coordinates
- Apply the formulas \(x = r\cos\theta\), \(y = r\sin\theta\), \(r = \sqrt{x^2 + y^2}\), and \(\tan\theta = y/x\)
- Identify when arctan gives a quadrant-incorrect angle and adjust appropriately
- Convert equations between rectangular and polar forms
- Recognize which coordinate system makes a given shape simpler to describe
Quick Check
Convert the rectangular point \((0, -5)\) to polar coordinates with \(\theta \in [0, 2\pi)\).
Instructor Notes
Teaching Notes
This simulation bridges the gap between rectangular and polar coordinates. The neighborhood metaphor grounds the abstraction in everyday experience before introducing formal notation. The right triangle derivation in Step 3 connects directly to Simulation 7's trigonometric definitions, reinforcing that these are the same ratios students have already used.
The arctan trap in Step 4 is a common source of lost points on exams. Emphasize that \(\arctan\) only returns values in \((-\pi/2, \pi/2)\), so a quadrant check is always required. The calculator cannot determine the quadrant from \(y/x\) alone because the signs cancel.
Common Student Errors
- Trusting \(\arctan(y/x)\) without checking the quadrant -- the most common error on polar conversion problems
- Confusing \(x = r\cos\theta\) with \(x = r\sin\theta\) (mixing up which trig function goes with which coordinate)
- Believing \(r\) can never be negative -- it can in some conventions, and some textbooks use negative \(r\)
- Forgetting to use \(\cos^2\theta + \sin^2\theta = 1\) when simplifying equation conversions
- Writing \(r = \pm 5\) instead of \(r = 5\) when converting \(x^2 + y^2 = 25\)
Discussion Questions
- Why does a circle centered at the origin have such a simple polar equation? What property of polar coordinates makes this happen?
- Can you think of a shape that is simpler in rectangular form than polar form? What makes it simpler?
- Why does \(\arctan\) lose quadrant information? What is it about the tangent function that causes this?
Exam Connection
Conversion problems appear frequently on exams. Students should practice both directions (rectangular to polar and polar to rectangular) and be especially careful with the quadrant check. Equation conversions are common exam problems -- test both the mechanical substitution and the geometric interpretation ("What shape is this?").