Polar Coordinates Calculator

The mathematical relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) relies on foundational trigonometry and the Pythagorean theorem. To convert from polar to Cartesian coordinates, we use the radius (r) and the angle (θ) measured in radians or degrees: x = r × cos(θ) and y = r × sin(θ). Conversely, to convert from Cartesian to polar coordinates, we determine the radius using the hypotenuse formula r = √(x² + y²) and the angle using the four-quadrant inverse tangent function θ = atan2(y, x).

Let us walk through converting a polar point where the radius (r) equals 4 and the angle (θ) equals π/3 radians (60 degrees) into Cartesian coordinates. First, compute the x-coordinate using the cosine function: x = 4 × cos(π/3). Since cos(π/3) = 0.5, x = 4 × 0.5 = 2. Next, compute the y-coordinate using the sine function: y = 4 × sin(π/3). Since sin(π/3) ≈ 0.866, y = 4 × 0.866 = 3.464. Thus, the polar point (4, π/3) corresponds to the Cartesian point (2, 3.464).

Always verify whether your calculator is set to degrees or radians before entering angle values to avoid severe calculation discrepancies. When converting from Cartesian to polar using inverse tangent, ensure you use the atan2 function rather than a standard arctan function, as atan2 correctly accounts for the specific quadrant in which the (x, y) point lies.