# Convert Polar to Rectangular Coordinates and Vice Versa

 The rectangular coordinates (x , y) and polar coordinates (R , t) are related as follows. y = R sin t    and    x = R cos t R 2 = x 2 + y 2    and    tan t = y / x To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). Problem 1: Convert the polar coordinates (5 , 2.01) and (0.2 , 53 o) to rectangular coordinates to three decimal places. Solution to Problem 1: For the first point (5 , 2.01) R = 5 and t = 2.01 and is in radians. Set your calculator to radians and use the above formulas for x and y in terms of R and t to obtain: x = R cos t = 5 cos 2.01 = -2.126 y = R sin t = 5 sin 2.01 = 4.525 For the second point (0.2 , 53 o) R = 0.2 and t = 53 o and is in degrees. Set your calculator to degrees and use the above formulas for x and y in terms of R and t to obtain: x = R cos t = 0.2 cos 53 = 0.120 y = R sin t = 0.2 sin 53 = 0.160 Problem 2: Convert the rectangular coordinates (1 , 1) and (-2 ,-4) to polar coordinates to three decimal places. Express the polar angle t in degrees and radians. Solution to Problem 2: We first find R using the formula R = sqrt [x 2 + y 2] for the point (1 , 1). R = sqrt [x 2 + y 2] = sqrt [1 + 1] = sqrt ( 2 ) We now find tan t using the formula tan t = y / x. tan t = 1 / 1 Using the arctan function of the calculator, we obtain. t = Pi / 4 or t = 45 o Point (1 , 1) in rectangular coordinates may be written in polar for as follows. ( sqrt ( 2 ) , Pi / 4 ) or ( sqrt ( 2 ) , 45 o ) Let us find find R using for the point (-2 , -4). R = sqrt [x 2 + y 2] = sqrt [4 + 16] = sqrt ( 20 ) = 2 sqrt ( 5 ) We now find tan t. tan t = - 4 / - 2 = 2 Using the arctan function of the calculator, we obtain. t = 1.107 or t = 63.435 o BUT since the rectangular coordinates x and y are both negative, the point is in quadrant III and we need to add Pi or 180 o to the value of t given by the calculator. Hence the polar angle t is given by t = 4.249 or t = 243.435 o Point (-2 , -4) in rectangular coordinates may be written in polar for as follows. ( 2 sqrt ( 5 ) , 4.249 ) or ( 2 sqrt ( 5 ) , 243.435 o ) More references on polar coordinates and trigonometry topics.