Polar Coordinates, Combined

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 to rectangular coordinates?

Back

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the formulas: x = r * cos(θ) and y = r * sin(θ).

Front

What is the formula for calculating the distance from the origin in polar coordinates?

Back

The distance from the origin in polar coordinates is given by the radius 'r'.

Front

What is the relationship between the angle θ and the coordinates in polar coordinates?

Back

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

Front

If r = 2 and θ = 75°, what are the rectangular coordinates?

Back

The rectangular coordinates are approximately (0.517, 1.931).

Front

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

Back

The angle θ determines the direction of the point in the polar coordinate system.

Front

How do you find the angle θ given rectangular coordinates (x, y)?

Back

To find θ, use the formula θ = arctan(y/x).

Front

What is the polar coordinate representation of the point (4, 7)?

Back

The polar coordinates can be found using r = √(x² + y²) and θ = arctan(y/x). For (4, 7), r ≈ 8.06 and θ ≈ 60.26°.

Front

What is the polar coordinate of the point (5, 5)?

Back

The polar coordinates are approximately (7.07, 45°).

Front

How do you determine if two points in polar coordinates are the same?

Back

Two points (r1, θ1) and (r2, θ2) are the same if r1 = r2 and θ1 = θ2, or if θ1 and θ2 differ by a multiple of 360°.