Given the polar coordinates left-parenthesis 4 comma negative Start Fraction pi over 3 End Fraction right-parenthesis, which of the following rectangular coordinate pairs represents the same point?
To convert from polar coordinates to rectangular coordinates, we use the formulas:
x = r*cos(theta)
y = r*sin(theta)
Given that r = 4 and theta = -pi/3, we have:
x = 4*cos(-pi/3)
y = 4*sin(-pi/3)
cos(-pi/3) = 1/2
sin(-pi/3) = -sqrt(3)/2
Therefore, the rectangular coordinates are:
(4 * 1/2, 4 * -sqrt(3)/2)
(2, -2sqrt(3))
The correct rectangular coordinate pair that represents the same point in the plane as the polar coordinates (4, -pi/3) is (2, -2sqrt(3)).