Final answer:
The point (4, 4) in polar coordinates is (5.66 m, π / 4 radians) or (5.66 m, 45°). Another set of coordinates for the same point, taking into account the periodicity of angles in polar coordinates, is (5.66 m, 9π / 4 radians) or (5.66 m, 405°).
Step-by-step explanation:
To determine two pairs of polar coordinates for the point (4, 4), we first need to understand the relationship between Cartesian coordinates and polar coordinates. The Cartesian coordinate (x, y) is equivalent to the polar coordinate (r, θ), where r is the radial distance from the origin to the point, and θ is the angle formed with the positive x-axis.
For the Cartesian point (4, 4), we calculate the radial distance using the Pythagorean theorem which gives us:
r = √(x² + y²) = √(4² + 4²) = √32 ≈ 5.66 m
The angle θ, in radians, can be found using the arctangent function:
θ = arctan(y / x) = arctan(4 / 4) = π / 4 or 45°
The principle angle is 45° (π / 4 radians). To find another set of polar coordinates for the same point, we can add 360° (or 2π radians) to the angle to obtain:
θ = 45° + 360° = 405°
Alternatively, in radians:
θ = π / 4 + 2π = 9π / 4
Therefore, two pairs of polar coordinates for the point (4, 4) are:
(5.66 m, 45°) or (5.66 m, π / 4 radians) and
(5.66 m, 405°) or (5.66 m, 9π / 4 radians).