Polar to Rectangular Coordinates Calculator

Polar coordinates represent a point in a plane using a distance from the origin, known as the radius (r), and an angle from the positive x-axis, known as theta (θ). To convert these values into rectangular coordinates (x, y), we utilize fundamental trigonometric identities. The horizontal component is computed using the formula x = r × cos(θ), while the vertical component is determined by y = r × sin(θ). Conversely, if you are converting rectangular coordinates back into polar form, you apply the Pythagorean theorem r = √(x² + y²) and the two-argument arctangent function θ = atan2(y, x).

Let us convert the polar coordinates (r = 10, θ = π/3 radians) into rectangular coordinates. First, find the x-coordinate by multiplying the radius by the cosine of the angle: x = 10 × cos(π/3). Since cos(π/3) equals 0.5, we get x = 10 × 0.5 = 5. Next, find the y-coordinate by multiplying the radius by the sine of the angle: y = 10 × sin(π/3). Since sin(π/3) equals approximately 0.866, we calculate y = 10 × 0.866 = 8.66. Thus, the rectangular coordinates are (5, 8.66).

Always double-check whether your calculator or software is set to degrees or radians before inputting angle values. Mixing up these two units is the most common cause of conversion errors. Additionally, pay close attention to the quadrant in which your point lies, as trigonometric signs for sine and cosine will naturally shift depending on whether the angle is in the first, second, third, or fourth quadrant.