Polar Coordinates: Angles and Distances Instead of x and y

Standard coordinates: (x, y). How far right, how far up.

Polar coordinates: (r, θ). How far out, what angle.

A point is described by its distance from the origin and its angle from the positive x-axis.

Same point, different language. And for some curves, the polar language is simpler.

The Unlock: Distance and Angle Instead of x and y

Cartesian coordinates (x, y) describe position as horizontal and vertical displacements.

Polar coordinates (r, θ) describe position as a radius and an angle.

r is the distance from the origin. θ is the angle (measured counterclockwise from the positive x-axis).

For problems with rotational symmetry—circles, spirals, orbits—polar coordinates are natural.

Conversion Between Polar and Cartesian

To convert from polar (r, θ) to Cartesian (x, y):

x = r cos(θ) y = r sin(θ)

Example: (r, θ) = (5, π/3).

x = 5 cos(π/3) = 5 · (1/2) = 2.5. y = 5 sin(π/3) = 5 · (√3/2) ≈ 4.33.

Cartesian coordinates: (2.5, 4.33).

To convert from Cartesian (x, y) to polar (r, θ):

r = √(x² + y²) θ = arctan(y/x) (adjust for quadrant)

Example: (x, y) = (3, 4).

r = √(9 + 16) = 5. θ = arctan(4/3) ≈ 0.93 radians ≈ 53°.

Polar coordinates: (5, 0.93).

Non-Uniqueness of Polar Coordinates

In Cartesian coordinates, each point has a unique (x, y).

In polar coordinates, points are not unique.

(r, θ) and (r, θ + 2π) represent the same point. Adding a full rotation doesn't change the location.

Also, (-r, θ) and (r, θ + π) represent the same point. Negative radius means you move in the opposite direction.

Example: (2, π/4) and (2, π/4 + 2π) = (2, 9π/4) are the same point.

Also, (-2, π/4) and (2, π/4 + π) = (2, 5π/4) are the same point.

This non-uniqueness is a quirk of polar coordinates.

The Pole and the Polar Axis

The pole is the origin (r = 0).

The polar axis is the positive x-axis (θ = 0).

These are the reference points for polar coordinates.

Polar Equations

A polar equation relates r and θ.

Example: r = 2.

This is the set of all points at distance 2 from the origin. A circle of radius 2.

Example: θ = π/4.

This is the set of all points at a 45° angle from the positive x-axis. A line through the origin.

Example: r = θ.

As θ increases, r increases. This is the Archimedean spiral. It spirals outward as you rotate.

Circles in Polar Coordinates

A circle centered at the origin with radius a:

r = a.

Simple and clean.

A circle passing through the origin, tangent to the y-axis:

r = 2a sin(θ).

A circle passing through the origin, tangent to the x-axis:

r = 2a cos(θ).

These are much simpler in polar form than in Cartesian.

Lines in Polar Coordinates

A line through the origin:

θ = constant.

Example: θ = π/3 is a line at 60° from the x-axis.

A vertical line x = a:

r cos(θ) = a, or r = a / cos(θ) = a sec(θ).

A horizontal line y = b:

r sin(θ) = b, or r = b / sin(θ) = b csc(θ).

Lines not through the origin are messier in polar form.

Rose Curves

r = a sin(nθ) or r = a cos(nθ).

These are rose curves. They have petals.

• If n is odd, the rose has n petals.
• If n is even, the rose has 2n petals.

Example: r = 3 sin(2θ).

n = 2 (even), so 4 petals. Maximum radius 3.

Example: r = 2 cos(5θ).

n = 5 (odd), so 5 petals. Maximum radius 2.

Rose curves are beautiful and symmetric.

Limaçons

r = a ± b cos(θ) or r = a ± b sin(θ).

These are limaçons (French for "snails").

The shape depends on the ratio a/b.

• If a/b < 1: limaçon with an inner loop.
• If a/b = 1: cardioid (heart shape).
• If 1 < a/b < 2: dimpled limaçon.
• If a/b ≥ 2: convex limaçon.

Example: r = 1 + cos(θ).

a = b = 1, so this is a cardioid.

Example: r = 2 + 3 sin(θ).

a = 2, b = 3, so a/b = 2/3 < 1. This has an inner loop.

Lemniscates

r² = a² cos(2θ) or r² = a² sin(2θ).

These are lemniscates (figure-eight curves).

Example: r² = 4 cos(2θ).

The curve has two loops, symmetric about the origin.

Lemniscates look like infinity symbols (∞).

Spirals

Archimedean spiral: r = aθ.

The radius increases linearly with the angle. The spiral has constant spacing between turns.

Logarithmic spiral: r = ae^(bθ).

The radius grows exponentially. This spiral appears in nautilus shells, galaxies, and hurricanes.

Hyperbolic spiral: r = a/θ.

As θ increases, r decreases. The spiral winds inward toward the pole.

Spirals are natural in polar coordinates.

Symmetry in Polar Coordinates

Symmetry about the x-axis: If (r, θ) is on the curve, so is (r, -θ).

Test: Replace θ with -θ. If the equation is unchanged, it's symmetric about the x-axis.

Symmetry about the y-axis: If (r, θ) is on the curve, so is (r, π - θ).

Test: Replace θ with π - θ.

Symmetry about the origin: If (r, θ) is on the curve, so is (-r, θ) or (r, θ + π).

Test: Replace r with -r or θ with θ + π.

Symmetry simplifies graphing. You only need to plot part of the curve and reflect.

Area in Polar Coordinates

The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is:

A = (1/2) ∫[α to β] r² dθ.

Example: Area of the circle r = a from θ = 0 to θ = 2π.

A = (1/2) ∫[0 to 2π] a² dθ = (1/2) a² · 2π = πa².

Correct—it's the area of a circle.

Example: Area of one petal of r = sin(2θ).

One petal spans from θ = 0 to θ = π/2.

A = (1/2) ∫[0 to π/2] sin²(2θ) dθ = π/8.

Polar integrals are natural for regions with rotational symmetry.

Arc Length in Polar Coordinates

The arc length of a polar curve r = f(θ) from θ = α to θ = β is:

L = ∫[α to β] √(r² + (dr/dθ)²) dθ.

Example: Arc length of r = θ from θ = 0 to θ = 2π (one turn of the Archimedean spiral).

dr/dθ = 1.

L = ∫[0 to 2π] √(θ² + 1) dθ.

This integral can be evaluated numerically or with a substitution.

Tangent Lines in Polar Coordinates

The slope of the tangent line in Cartesian coordinates is dy/dx.

In polar coordinates:

dy/dx = (dr/dθ · sin(θ) + r cos(θ)) / (dr/dθ · cos(θ) - r sin(θ)).

This comes from differentiating x = r cos(θ), y = r sin(θ).

Example: r = 1 + cos(θ) at θ = π/2.

r = 1 + cos(π/2) = 1.

dr/dθ = -sin(θ), so dr/dθ|_(π/2) = -1.

dy/dx = ((-1) · 1 + 1 · 0) / ((-1) · 0 - 1 · 1) = -1 / -1 = 1.

The tangent line has slope 1.

Why Polar Coordinates?

Some curves are simpler in polar form.

Cartesian: x² + y² = 4 (circle). Polar: r = 2.

Cartesian: (x² + y²)² = x² - y² (lemniscate). Polar: r² = cos(2θ).

For curves with rotational symmetry, polar is cleaner.

Polar Coordinates in Physics

Orbital mechanics: Planets move in ellipses. In polar coordinates with the sun at the origin, the orbit is:

r = a(1 - e²) / (1 + e cos(θ)).

Simple and elegant.

Angular motion: Rotating objects are naturally described in polar coordinates. Angular velocity, angular momentum, torque—all use polar or cylindrical coordinates.

Wave propagation: Circular and spherical waves are simplest in polar (2D) or spherical (3D) coordinates.

Polar coordinates are the language of rotation.

Polar Equations for Conics

All conic sections can be written in polar form with one focus at the origin:

r = ed / (1 + e cos(θ)).

• e = eccentricity.
• d = distance from focus to directrix.

For e < 1: ellipse. For e = 1: parabola. For e > 1: hyperbola.

This unified form is beautiful.

Example: Earth's orbit (e ≈ 0.017, nearly circular):

r ≈ a(1 - e²) / (1 + e cos(θ)) ≈ constant.

Graphing Polar Curves

To graph r = f(θ):

1. Make a table: choose values of θ, compute r.
2. Plot points (r, θ) in polar coordinates.
3. Connect the points smoothly.
4. Use symmetry to reduce the work.

Example: r = 2 + 2 cos(θ) (cardioid).

θ = 0: r = 4. θ = π/2: r = 2. θ = π: r = 0. θ = 3π/2: r = 2. θ = 2π: r = 4.

The curve is symmetric about the x-axis. Plot the top half, reflect for the bottom.

Converting Equations Between Cartesian and Polar

Sometimes you want to convert an equation from one form to the other.

Polar to Cartesian: Use x = r cos(θ), y = r sin(θ), r² = x² + y².

Example: r = 4 sin(θ).

Multiply both sides by r: r² = 4r sin(θ).

x² + y² = 4y.

This is a circle in Cartesian form.

Cartesian to Polar: Use r = √(x² + y²), tan(θ) = y/x.

Example: x² + y² = 9.

r² = 9, so r = 3.

A circle in polar form.

Negative Radius

In polar coordinates, r can be negative.

(r, θ) with r < 0 means: move in the direction θ + π (opposite direction) a distance |r|.

Example: (-2, 0) is the same as (2, π). Both represent the point (-2, 0) in Cartesian coordinates.

Negative radius is a quirk, but sometimes it simplifies equations.

Polar Coordinates in Three Dimensions

Polar coordinates extend to 3D as cylindrical coordinates: (r, θ, z).

r and θ describe position in the xy-plane. z is the height.

And spherical coordinates: (ρ, θ, φ).

ρ is the distance from the origin. θ is the azimuthal angle. φ is the polar angle.

These are essential in physics and engineering.

Why Polar Coordinates Are in Precalculus

Polar coordinates introduce a new way to describe position.

They connect trigonometry (sine, cosine) to geometry (circles, spirals).

They prepare you for vector calculus, where you'll work with different coordinate systems.

And they show that there's no single "right" way to describe space. Different problems call for different coordinates.

Common Mistakes

Mistake 1: Forgetting to adjust for quadrant when converting to polar.

arctan(y/x) only gives angles in (-π/2, π/2). If the point is in quadrant II or III, you need to add π.

Mistake 2: Treating polar coordinates as unique.

(r, θ) and (r, θ + 2πn) represent the same point. Multiple representations exist.

Mistake 3: Ignoring negative radius.

(-r, θ) is valid and equals (r, θ + π).

Mistake 4: Mixing up r and θ in polar equations.

r is the dependent variable (output). θ is the independent variable (input). Don't confuse them.

Mistake 5: Assuming all curves are simpler in polar.

Lines and rectangles are messier in polar. Circles and spirals are simpler. Choose the right tool.

The Payoff: Matching the Coordinate System to the Problem

Cartesian coordinates are great for horizontal and vertical motion.

Polar coordinates are great for rotation and radial motion.

When you understand both, you can choose the coordinate system that makes the problem simple.

That's the real skill: recognizing which representation clarifies the structure.

Coordinates aren't neutral. They shape how you see the problem. Choose wisely.

Part 9 of the Precalculus series.

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