How do you convert the polar coordinate ( 4.5541 , 1.2352 ) into cartesian coordinates?

1 Answer
Jan 29, 2016

Polar coordinates are in the form (r,θ), Cartesian (also called rectangular) coordinates are in the form (x,y). In this case the Cartesian coordinates are (1.50,4.30).

Explanation:

Polar coordinates are stated in the form (r,θ), a distance from the origin and an angle, in radians, counterclockwise from the positive x axis. In this case, we have a point 4.5541 units from the origin, at an angle of 1.2352 rad.

Cartesian coordinates, sometimes also called 'rectangular coordinates', are expressed in the form (x,y) where x is the distance along the x axis and y is the distance up the y axis.

To convert from one to the other, we use trigonometry. The r value from the polar coordinates is the hypotenuse of a right-angled triangle, and the x and y coordinates are the adjacent and opposite sides respectively, from the perspective of the angle θ.

Knowing the angle and a side allows us to use the definitions of sine and cosine to find the x and y coordinates:

sinθ=oppositehypotenuse opposite=hypotenusesinθ

In this case, y=4.5541sin(1.2352)=4.55410.9442=4.30

(Be sure to ensure that your calculator is on the 'radians', not 'degrees' setting when calculating the sine and cosine in this context.)

By similar reasoning:

cosθ=adjacenthypotenuse adjacent=hypotenusecosθ

In this case, x=4.5541cos(1.2352)=4.55410.3293=1.50

So that you don't have to remember all this discussion every time, you can memorize (but should understand):

x=rcosθ

y=rsinθ

Combine these, and the Cartesian coordinates for the point are (1.50,4.30).