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 origin) and an angle from a reference direction.

Front

What is the relationship between polar coordinates and Cartesian coordinates?

Back

Polar coordinates (r, θ) can be converted to Cartesian coordinates (x, y) using the formulas: x = r * cos(θ) and y = r * sin(θ).

Front

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

Back

To convert (x, y) to polar coordinates, use: r = √(x² + y²) and θ = arctan(y/x).

Front

What is the angle θ in polar coordinates measured from?

Back

The angle θ in polar coordinates is measured counterclockwise from the positive x-axis.

Front

What does the 'r' represent in polar coordinates?

Back

In polar coordinates, 'r' represents the radial distance from the origin to the point.

Front

What is the polar coordinate for the point (2, 0°)?

Back

The polar coordinate (2, 0°) represents a point 2 units away from the origin along the positive x-axis.

Front

How do you find the polar coordinates of a point given in Cartesian coordinates (3, 4)?

Back

For (3, 4), r = √(3² + 4²) = 5 and θ = arctan(4/3) ≈ 53.13°. Thus, polar coordinates are (5, 53.13°).

Front

What is the polar coordinate for the point at (0, 0)?

Back

The polar coordinate for the point at (0, 0) is (0, θ) for any angle θ, since it is at the origin.

Front

What is the significance of the angle being negative in polar coordinates?

Back

A negative angle in polar coordinates indicates a clockwise rotation from the positive x-axis.

Front

How do you express the point (4, π/3) in Cartesian coordinates?

Back

For (4, π/3), x = 4 * cos(π/3) = 2 and y = 4 * sin(π/3) = 2√3. Thus, Cartesian coordinates are (2, 2√3).