Polar Coordinates

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.

Front

How is an ordered pair written in polar form?

Back

An ordered pair in polar form is written as (r, θ), where r is the radial distance and θ is the angle.

Front

What is the relationship between rectangular coordinates (x, y) and polar coordinates (r, θ)?

Back

The relationship is given by the equations: r = √(x² + y²) and θ = arctan(y/x).

Front

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

Back

To convert, calculate r = √(x² + y²) and θ = arctan(y/x).

Front

What is the polar coordinate representation of the point (0, 0)?

Back

The polar coordinate representation of the point (0, 0) is (0, θ) for any angle θ.

Front

What does the angle θ represent in polar coordinates?

Back

The angle θ represents the direction of the point from the positive x-axis, measured in degrees or radians.

Front

What is the significance of the radial distance r in polar coordinates?

Back

The radial distance r indicates how far the point is from the origin (pole) in the polar coordinate system.

Front

How do you find the radial distance r given rectangular coordinates x = 12 and y = 5?

Back

Use the formula r = √(x² + y²). Here, r = √(12² + 5²) = √(144 + 25) = √169 = 13.

Front

Convert the polar coordinates (4, 45°) to rectangular coordinates.

Back

To convert, use x = r * cos(θ) and y = r * sin(θ). Thus, x = 4 * cos(45°) = 4 * √2/2 = 2√2 and y = 4 * sin(45°) = 4 * √2/2 = 2√2.

Front

What is the polar coordinate of the point that is 4 units away from the origin at an angle of 225°?

Back

The polar coordinate is (4, 225°).