Subjects

AP Calculus AB/BC

Dec 9, 2025

2 pages

Understanding Polar Graphs: Basics and Applications

Kate Bicalho @katebicalho_ipsn

Polar coordinates offer a different way to represent points in a plane, using a distance from the origin... Show more

NOTE
Polar graphs
Coordinates and graphs
consists of a fixed point (r.)
polar coordinates are not unique!
(r, 0) =(-r, 0+)
x=rcose
yr sine
r

Polar Coordinates and Graphs

Polar coordinates describe a point using the notation (r,θ), where r is the distance from the origin and θ is the angle from the positive x-axis. An interesting property is that polar coordinates aren't unique - the point (r,θ) is identical to $-r,θ+π$ .

Converting between rectangular (x,y) and polar coordinates is straightforward

x = r cos θ
y = r sin θ
r = √ $x²+y²$
θ = tan⁻¹ $y/x$

Finding the slope of a tangent line to a polar curve requires using the chain rule. For a curve r = f(θ), the derivative is

dy/dx = [f(θ)cos θ + f'(θ)sin θ]/[-f(θ)sin θ + f'(θ)cos θ]

Pro Tip When looking for horizontal or vertical tangents, set dy/dθ = 0 for horizontal tangents and dx/dθ = 0 for vertical tangents instead of working with the full derivative formula.

Area in Polar Coordinates

Calculating areas using polar coordinates is often simpler than using rectangular coordinates for certain shapes. The formula for the area bounded by a polar curve r = f(θ) from θ = α to θ = β is

A = (1/2)∫ $α to β$ $f(θ)$ ² dθ or more simply A = (1/2)∫ $α to β$ r² dθ

When working with symmetric curves, you can often calculate the area for just one portion and then multiply by the number of symmetric parts. This trick can save considerable calculation time.

For integrals involving cos²(nθ), remember the identity cos²x = $1+cos(2x)$ /2, which can simplify your work significantly. This approach converts complicated trigonometric integrals into more manageable forms.

Remember When a polar curve has symmetry, you can integrate over a smaller interval and multiply by the number of symmetric sections to find the total area.

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Subjects

AP Calculus AB/BC

•

Dec 9, 2025

•

2 pages

Understanding Polar Graphs: Basics and Applications

Kate Bicalho

@katebicalho_ipsn

Polar coordinates offer a different way to represent points in a plane, using a distance from the origin (r) and an angle (θ) instead of x and y coordinates. This system is particularly useful for describing circular or spiral patterns... Show more

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Polar Coordinates and Graphs

Converting between rectangular (x,y) and polar coordinates is straightforward:

x = r cos θ
y = r sin θ
r = √ $x²+y²$
θ = tan⁻¹ $y/x$

Finding the slope of a tangent line to a polar curve requires using the chain rule. For a curve r = f(θ), the derivative is:

dy/dx = [f(θ)cos θ + f'(θ)sin θ]/[-f(θ)sin θ + f'(θ)cos θ]

Pro Tip: When looking for horizontal or vertical tangents, set dy/dθ = 0 for horizontal tangents and dx/dθ = 0 for vertical tangents instead of working with the full derivative formula.

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Area in Polar Coordinates

A = (1/2)∫ $α to β$ $f(θ)$ ² dθ or more simply A = (1/2)∫ $α to β$ r² dθ

When working with symmetric curves, you can often calculate the area for just one portion and then multiply by the number of symmetric parts. This trick can save considerable calculation time.

Remember: When a polar curve has symmetry, you can integrate over a smaller interval and multiply by the number of symmetric sections to find the total area.