Polar coordinates are stated in the form (r, theta)(r,θ), a distance from the origin and an angle, in radians, counterclockwise from the positive xx axis. In this case, we have a point 4.55414.5541 units from the origin, at an angle of 1.23521.2352 radrad.
Cartesian coordinates, sometimes also called 'rectangular coordinates', are expressed in the form (x, y)(x,y) where xx is the distance along the xx axis and yy is the distance up the yy axis.
To convert from one to the other, we use trigonometry. The rr value from the polar coordinates is the hypotenuse of a right-angled triangle, and the xx and yy coordinates are the adjacent and opposite sides respectively, from the perspective of the angle thetaθ.
Knowing the angle and a side allows us to use the definitions of sine and cosine to find the xx and yy coordinates:
sin theta = (opposite)/("hypotenuse") tosinθ=oppositehypotenuse→ opposite = "hypotenuse"* sin thetaopposite=hypotenuse⋅sinθ
In this case, y = 4.5541*sin(1.2352) = 4.5541*0.9442=4.30y=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") tocosθ=adjacenthypotenuse→ adjacent = "hypotenuse"* cos thetaadjacent=hypotenuse⋅cosθ
In this case, x = 4.5541*cos(1.2352) = 4.5541*0.3293=1.50x=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 thetax=rcosθ
y=rsinthetay=rsinθ
Combine these, and the Cartesian coordinates for the point are (1.50, 4.30)(1.50,4.30).
