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# How to Add Polar Coordinates?

TopPoint in maths can be drawn by means of 2 illustrations viz. Cartesian and polar coordinate system. Polar coordinate system is referred to angle that states the direction of vector and a specific distance from a fixed Point that signifies the size of the vector. Adding rectangular coordinates means to add x and y- coordinates separately. Same procedure is used in case of Polar Coordinates also. Let us learn how to add polar coordinates.

Suppose there are two polar coordinates given as: A (4 cos 30, 4 sin 30) and B (2 cos 60, M sin 60). To add these coordinates, first we convert them to their respective rectangular coordinates as follows:

First polar coordinate has radial distance as: 4 and that for second polar coordinate is: 2. There corresponding rectangular coordinates can be found by following formula:
X1 = 4 cos 30,
Y1 = 4 sin 30,
And
X2 = 2 cos 60,
Y2 = 2 sin 60,
We get rectangular coordinates as: (2√3, 2) and (1, √3). On adding rectangular coordinates we get new coordinates as: (2√3 + 1, 2 + √3) = (1 + 2√3, 2 + √3).

To represent these coordinates again as polar coordinates, we use same formula:
1 + 2√3 = r cos a,
2 + √3 = r sin a,
Where, r = ((1 + 2√3)2 + (2 + √3)2)1 /2 = (1 + 12 + 4√3 + 4 + 3 + 4√3)1 /2 = (20 + 8√3)1 /2 =2 √(5 + 2√3).
And a = tan-1 (2 + √3) / (1 + 2√3),
So, resultant polar coordinates are: (2 √(5 + 2√3) cos (tan-1 (2 + √3) / (1 + 2√3)), 2 √(5 + 2√3) sin (tan-1 (2 + √3) / (1 + 2√3))).