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

1 Answer
Jan 29, 2016

Answer:

Polar coordinates are in the form #(r, theta)#, 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, theta)#, 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 #theta#.

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

#sin theta = (opposite)/("hypotenuse") to# #opposite = "hypotenuse"* sin theta#

In this case, #y = 4.5541*sin(1.2352) = 4.5541*0.9442=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:

#cos theta = (adjacent)/("hypotenuse") to# #adjacent = "hypotenuse"* cos theta#

In this case, #x = 4.5541*cos(1.2352) = 4.5541*0.3293=1.50#

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

#x = r cos theta#

#y=rsintheta#

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