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 , 2?) or [0 , 360°).

Examples on Converting Polar and Rectangular Coordinates

Example 1

Convert the polar coordinates (5 , 2.01) and (0.2 , 53°) to rectangular coordinates to three decimal places.
Solution to Example 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°), R = 0.2 and t = 53° 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

Example 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 Example 2

We first find R using the formula R = √ [x^{ 2} + y^{ 2}] for the point (1 , 1).
R = √ [x^{ 2} + y^{ 2}] = √ [1 + 1] = √ ( 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.
( √ ( 2 ) , ? / 4 ) or ( √ ( 2 ) , 45 ^{ o} )

Let us find find R using for the point (-2 , -4).
R = √ [x^{ 2} + y^{ 2}] = √ [4 + 16] = √ ( 20 ) = 2 √ ( 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 ? or 180° to the value of t given by the calculator. Hence the polar angle t is given by
t = 4.249 or t = 243.435°

Point (-2 , -4) in rectangular coordinates may be written in polar for as follows.
( 2 √ ( 5 ) , 4.249 ) or ( 2 √ ( 5 ) , 243.435° )

More References and Links to Polar Coordinates and Trigonometry