Polar Coordinates and Graphs

Front

What are polar coordinates?

Back

Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance from a reference point (the pole) and an angle from a reference direction (the polar axis).

Front

How do you convert polar coordinates (r, θ) to rectangular coordinates (x, y)?

Back

The conversion formulas are: x = r * cos(θ) and y = r * sin(θ).

Front

What is the ordered pair in polar coordinates?

Back

The ordered pair in polar coordinates is written as (r, θ).

Front

What is the polar equation of a circle with radius 3 centered at the origin?

Back

The polar equation is r = 3.

Front

What is a cardioid in polar coordinates?

Back

A cardioid is a heart-shaped curve represented by the polar equation r = a + b * sin(θ) or r = a + b * cos(θ), where a and b are constants.

Front

How do you find the area enclosed by a polar curve?

Back

The area A enclosed by a polar curve r = f(θ) from θ = a to θ = b is given by the formula: A = 1/2 * ∫[a to b] (f(θ))^2 dθ.

Front

What is the significance of the angle θ in polar coordinates?

Back

The angle θ represents the direction of the point from the origin, measured counterclockwise from the positive x-axis.

Front

What is the polar equation for a limacon with an inner loop?

Back

The polar equation for a limacon with an inner loop is of the form r = a + b * sin(θ) or r = a + b * cos(θ) where |b| > |a|.

Front

How do you graph polar coordinates?

Back

To graph polar coordinates, plot the point at a distance r from the origin at an angle θ from the polar axis.

Front

What is the relationship between polar and rectangular coordinates?

Back

The relationship is defined by the equations: x = r * cos(θ) and y = r * sin(θ), and the inverse is r = √(x^2 + y^2) and θ = tan^(-1)(y/x).