Final answer:
To convert Cartesian coordinates (-2,2) to polar coordinates, calculate the radial distance (r = 2√2) and then determine the angle (θ = 3π/4 radians), considering it is in the second quadrant.
Step-by-step explanation:
To convert the Cartesian coordinates (-2,2) into equivalent polar coordinates, we need to find the radial coordinate (r) and the angle (θ). The radial coordinate r is the distance from the origin (0,0) to the point (-2,2). We calculate this using the Pythagorean theorem: r = √((-2)^2 + (2)^2) = √(4+4) = √8 = 2√2.
To find the angle θ, we use the arctangent function (tan⁻¹ or atan), which gives the angle whose tangent is the ratio of the y-coordinate to the x-coordinate. Since we are in the second quadrant where x is negative and y is positive, θ will be between 90° and 180°, or π/2 and π radians. Using the point (-2,2), we have θ = tan⁻¹(2/(-2)) = tan⁻¹(-1). Tan⁻¹(-1) gives -45° or -π/4 radians, but since we're in the second quadrant, we add π to find the correct angle: θ = π + (-π/4) = (3π/4) radians.
Thus, the point (-2,2) can be described in polar coordinates as (2√2, 3π/4).