A polar coordinate gives a distance from the origin and an angle from the positive x-axis.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Polar to Cartesian
x = r cos θ and y = r sin θ. Both come straight from the right triangle.
Cartesian to polar
r is the hypotenuse by Pythagoras, and θ comes from the arctangent — with the quadrant checked by hand.
A point has many names
Adding 360° to θ lands on the same point, and a negative r points the other way.
Distance and angle
A polar coordinate (r, θ) locates a point by distance from the origin and angle from the positive x-axis. It is the natural system for anything with rotational symmetry.
Converting between systems
x = r cos θ and y = r sin θ go one way; r = √(x² + y²) and θ = tan⁻¹(y/x) return, with quadrant adjustment. Both directions appear routinely in BC problems.
Points have many representations
Adding 2π to θ, or negating r and adding π, gives the same point. That non-uniqueness matters when finding intersections of polar curves, where two curves can meet at a point they reach at different θ.
Negative r is meaningful
A negative r places the point opposite the direction θ indicates. The convention keeps many curves continuous and is what produces the inner loops of limaçons.
Step 2: Try It Yourself
Tap and try it out.
- Angle60° = π/3 rad
- x-coordinate0.500
- y-coordinate0.866
- cos 60°0.500
- sin 60°0.866
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Omar Convert a Point
Omar converts the polar point r = 2, θ = 90° to Cartesian.
- Step 1
x = 2 cos 90°, and cos 90° is 0.
Step 4: Your Turn
Practice makes it stick.
Straight Out
Problem 1 of 2
r = 5, θ = 0°. What is x?
Straight Up
Problem 2 of 2
r = 5, θ = 90°. What is x?
Between the Systems
1 of 8
r = 3, θ = 0°. What is y?
2 of 8
r = 4, θ = 90°. What is y?
3 of 8
Cartesian (3, 4). What is r?
4 of 8
Cartesian (0, 7). What is r?
5 of 8
r = 6, θ = 180°. What is x?
6 of 8
Set the angle to 270°, where cosine is 0 and sine is −1.
- Angle0° = 0 rad
- x-coordinate1
- y-coordinate0
- cos 0°1
- sin 0°0
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
7 of 8
Does adding 360° to θ change the point? 1 for yes, 0 for no.
8 of 8
Cartesian (0, 0). What is r?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Cartesian (6, 8). What is r?
What You Learned
- A polar coordinate is a distance and an angle.
- x = r cos θ and y = r sin θ convert to Cartesian; Pythagoras converts back.
- One point has infinitely many polar names.