Polar Coordinates and Curves Free Online FlashCards

What do r and θ represent in polar coordinates?

A polar point is written (r, θ), where r is directed distance from the pole and θ is the counterclockwise angle from the positive x-axis.

How can one polar point have multiple representations?

The same point can be written (r, θ + 2πk) or (−r, θ + π + 2πk), where k is any integer.

What equation relates r, x, and y?

The rectangular-to-polar distance relationship is r² = x² + y².

What is the polar form of y = 2x?

For y = 2x, substitution gives tan θ = 2, so the line through the origin is θ = arctan(2).

How is symmetry about the polar axis tested?

Replace θ with −θ. If the equation is unchanged, the curve is symmetric about the polar axis, the x-axis.

What curve does r = a represent?

The equation r = a represents a circle centered at the pole with radius |a|.

How can a polar equation be converted to rectangular form?

For r = f(θ), use x = f(θ) cos θ and y = f(θ) sin θ, then eliminate θ if possible.

What is the slope formula for a polar curve?

dy/dx = [r′ sin θ + r cos θ] / [r′ cos θ − r sin θ], provided dx/dθ ≠ 0.

What is the tangent line to r = 1 + sin θ at θ = π/2?

At θ = π/2, r = 2 and dy/dθ = 0 while dx/dθ = −2, so the slope is 0 and the tangent line is y = 2.

How are horizontal and vertical polar tangents identified?

A horizontal tangent generally requires dy/dθ = 0 and dx/dθ ≠ 0; a vertical tangent generally requires dx/dθ = 0 and dy/dθ ≠ 0.

What is the polar area formula?

The area swept from θ = α to θ = β is A = ½∫α^β r² dθ, provided the interval traces the region appropriately.

What is the area of one petal of r = 3 sin(2θ)?

For r = 3 sin(2θ), one petal is traced on 0 ≤ θ ≤ π/2, and its area is 9π/8.