Final answer:
To convert Cartesian coordinates (7, -2) to polar coordinates, we calculate the radial coordinate r as √53 and the angle θ as atan2(-2, 7), which is approximately 5.998 radians. The polar coordinates are therefore roughly ( √53, 5.998 radians).
Step-by-step explanation:
To convert the Cartesian coordinates (7, − 2) to polar coordinates, we need to calculate the radial coordinate (r) and the angle (θ). The radial coordinate r is the distance from the origin to the point, which can be found using the Pythagorean theorem:
r = √(x² + y²) = √(7² + (-2)²) = √(49 + 4) = √53
The angle θ in radians is the arctangent of the ratio of y over x, which is:
θ = atan2(y, x) = atan2(-2, 7)
Since our point lies in the fourth quadrant (because x is positive and y is negative), we need to make sure our angle is measured from the positive x-axis in a counterclockwise direction. If we get a negative value for θ, we should add 2π to it to ensure it falls between 0 and 2π. Performing the calculation:
θ = atan2(-2, 7) ≈ -0.286 + 2π ≈ 5.998 (radians)
Lastly, we represent the polar coordinates of the point (7, − 2) as (r, θ). According to the computed values, the polar coordinates are approximately ( √53, 5.998 radians).