# Which set of polar coordinates describes the same location as the rectangular coordinates (4, -4)?

Which set of polar coordinates describes the same location as the rectangular coordinates (4, -4)?

A. (4, 45°)

B. (-4√2 , 315°)

C. (4√2 , 135°)

D. (4√2 , 315°)

• D.

The line from the origin to (4, -4) has length 4 sqrt 2 by Pythagoras' theorem, which rules out both A (wrong size) and B (not negative). And by convention the angle is measured around the origin anticlockwise from the positive x axis, hence it's 315 degrees.

• from P(x, y) to P(r, θ)

graph any point on the xy plane, , and draw a radius from the origin to the point, label it's length r, the angle between the x-axis and the radius you label θ. from trigonometry:

x = rcos(θ), y = rsin(θ), x^2 + y^2 = r^2 (you can write more relationships, but this will be sufficient to convert

r^2 = x^2 + y^2 = 16 + 16 = 32

r = sqrt( 32 ) = sqrt( 16 ) * sqrt( 2 ) = 4sqrt(2)

we could solve for θ as well using arctan(y/x), in this case your graph shows this angle is -45° or -pi/4

which is the same as 315°