Polar and Cartesian coordinates are two ways to describe the same location in a plane. Cartesian coordinates use horizontal and vertical distances, written as (x, y), while polar coordinates use a distance from the origin and an angle, written as (r, θ). Converting between them is important in calculus because many curves, areas, and motion problems are easier to describe in one system than the other.
Circles, spirals, and rotations often look simpler in polar form.
Key Facts
- Cartesian to polar distance: r = sqrt(x^2 + y^2)
- Polar to Cartesian horizontal coordinate: x = r cos θ
- Polar to Cartesian vertical coordinate: y = r sin θ
- Angle relation when x is not 0: tan θ = y / x
- Basic identity linking both systems: x^2 + y^2 = r^2
- For equations, replace x with r cos θ, y with r sin θ, and x^2 + y^2 with r^2
Vocabulary
- Cartesian coordinates
- A coordinate system that locates a point by its horizontal position x and vertical position y from the origin.
- Polar coordinates
- A coordinate system that locates a point by its distance r from the origin and its angle θ from the positive x-axis.
- Radius vector
- The directed segment from the origin to a point in polar coordinates.
- Polar angle
- The angle θ measured from the positive x-axis to the radius vector, usually counterclockwise as positive.
- Quadrant
- One of the four regions of the coordinate plane used to determine the correct signs of x and y and the correct angle θ.
Common Mistakes to Avoid
- Using tan θ = x / y instead of tan θ = y / x. This reverses the legs of the right triangle and usually gives the wrong angle.
- Forgetting the quadrant when finding θ from arctan(y / x). The calculator angle may have the right tangent value but point in the wrong direction.
- Treating r as always positive without checking the representation. A negative r points in the direction opposite the angle, so (r, θ) and (-r, θ + π) can represent the same point.
- Replacing r with x^2 + y^2 instead of replacing r^2 with x^2 + y^2. Since r = sqrt(x^2 + y^2), this error changes the equation.
Practice Questions
- 1 Convert the polar point (6, π/3) to Cartesian coordinates. Give exact values for x and y.
- 2 Convert the Cartesian point (-3, 3sqrt(3)) to polar coordinates with r > 0 and 0 ≤ θ < 2π.
- 3 Explain why the equation r = 4 cos θ represents a circle when converted to Cartesian form, and identify the circle's center and radius.