Polar Coordinates

Polar coordinates are a system for locating points in a plane using a distance from a fixed point (the pole) and an angle measured from a fixed line (the polar axis). In polar coordinates, a point is represented by its distance from the pole (r) and the angle it makes with the polar axis (θ).

Converting between Polar and Rectangular Coordinates

To convert from polar coordinates to rectangular coordinates, you can use the following formulas:

x = r * cos(θ)

y = r * sin(θ)

Where (x, y) are the rectangular coordinates, r is the distance from the pole, and θ is the angle with the polar axis.

To convert from rectangular coordinates to polar coordinates, you can use the following formulas:

r = √(x^2 + y^2)

θ = arctan(y / x)

Graphing in Polar Coordinates

When graphing in polar coordinates, you plot points based on their distance from the pole and the angle they make with the polar axis. The angle is measured in radians, with 2π radians being equivalent to 360 degrees.

A circle in polar coordinates is represented by the equation r = a, where 'a' is the radius of the circle. A line in polar coordinates is represented by the equation θ = b, where 'b' is the angle with the polar axis.

Practice Problems

1. Convert the point (3, π/4) from rectangular coordinates to polar coordinates.
2. Convert the point with polar coordinates (4, 60°) to rectangular coordinates.
3. Graph the polar equation r = 2cos(θ).

These are some of the basic concepts and problems related to polar coordinates. Practice more problems to gain confidence in this topic.

Hope this study guide helps!