Final answer:
To express the Cartesian coordinates (4,4) in polar coordinates, you can use the distance formula and the trigonometric functions. The polar coordinates can be expressed as (4(sqrt(2)), π/4) or (4(sqrt(2)), 3π/4).
Step-by-step explanation:
To express the Cartesian coordinates (4,4) in polar coordinates, we can use the distance formula and the trigonometric functions:
Step 1: Find the distance (r) from the origin to the point (4,4) using the distance formula:
r = sqrt((x^2) + (y^2)) = sqrt((4^2) + (4^2)) = sqrt(32) = 4(sqrt(2))
Step 2: Find the angle (θ) that the radial vector makes with the positive x-axis using the inverse tangent function:
θ = arctan(y / x) = arctan(4 / 4) = arctan(1) = π/4
Therefore, the polar coordinates of (4,4) can be expressed as (4(sqrt(2)), π/4).
Another way to express the coordinates is to use the negative angle:
θ = π - π/4 = 3π/4
So, the polar coordinates can also be expressed as (4(sqrt(2)), 3π/4).