Polar to Cartesian Coordinates Calculator

A polar coordinate is represented by a radius (r) and an angle (theta), expressed as (r, theta). The cartesian coordinate system uses standard horizontal and vertical axes, expressed as (x, y). To convert from polar to cartesian coordinates, we use fundamental right-triangle trigonometry. The horizontal component (x) is calculated using the cosine function: x = r * cos(theta). The vertical component (y) is found using the sine function: y = r * sin(theta). If your angle is given in degrees, it must first be converted to radians by multiplying by pi/180 before applying trigonometric functions.

Let us convert the polar coordinates (r = 5, theta = pi/3 radians) into cartesian coordinates. First, identify your values: radius r = 5 and angle theta = 60 degrees (pi/3). Apply the x-coordinate formula: x = 5 * cos(pi/3). Since the cosine of 60 degrees is 0.5, we get x = 5 * 0.5 = 2.5. Next, apply the y-coordinate formula: y = 5 * sin(pi/3). Since the sine of 60 degrees is approximately 0.866, we calculate y = 5 * 0.866 = 4.33. Thus, the resulting cartesian coordinates are approximately (2.5, 4.33).

Always double-check your calculator's angle mode before computing trigonometric values; mixing up degrees and radians is the most common source of calculation errors. Keep in mind that negative radius values simply reflect the point across the origin in the opposite direction of the specified angle. When graphing complex numbers or rotational forces, sketch a quick reference quadrant to verify that your resulting x and y signs match your intuitive expectations.