In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)

Verified step by step guidance

1. Step 1: Understand the relationship between rectangular and polar coordinates. Rectangular coordinates are given as (x, y), and polar coordinates are given as (r, \theta).
2. Step 2: Calculate the radial distance r using the formula r = \sqrt{x^2 + y^2}. For the point (5, 0), substitute x = 5 and y = 0 into the formula.
3. Step 3: Determine the angle \theta using the formula \theta = \arctan\left(\frac{y}{x}\right). Since y = 0, consider the angle on the x-axis.
4. Step 4: Express \theta in radians. Since the point lies on the positive x-axis, \theta is 0 radians.
5. Step 5: Combine the results to express the polar coordinates as (r, \theta).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point in a two-dimensional space using an ordered pair (x, y). The x-coordinate indicates the horizontal position, while the y-coordinate indicates the vertical position. Understanding how to interpret these coordinates is essential for converting them into polar coordinates.

Polar Coordinates

Polar coordinates describe a point in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The polar coordinates are expressed as (r, θ), where r is the radial distance and θ is the angle in radians. Converting from rectangular to polar coordinates involves calculating these two values based on the given x and y.

Angle Measurement in Radians

Radians are a unit of angular measurement where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. To convert degrees to radians, one can use the conversion factor π radians = 180 degrees. Understanding how to express angles in radians is crucial for accurately representing the angle θ in polar coordinates.