Polar Coordinates question?

Question

What is the relation of tan (theta) = y/x to quadrants on the polar plane?

I'm looking at a question where tan(theta)= (-1), so it follows that the polar coordinates are in quadrant 2 (the upper left quadrant)...why is this? What tan values are the other quadrants defined by? I know it probably has something to do with slope...

thanks

Answer 1

Tan θ is positive in the first and third quadrants and negative in the second and fourth quadrants.

For 0 ≤ θ ≤ 2π

tan θ = -1

θ = arctan(-1) = 3π/4 and 7π/4

Answer 2

tan(theta)=y/x
theta is the angle measured counterclockwise from the positive x-axis

quadrant 1 (upper right) theta=0 to 90 degrees tan(theta)=0 to oo
quadrant 2 (upper left) theta=90 to 180 degrees tan(theta)= -oo to 0
quadrant 3 (lower left) theta=180 to 270 degrees tan(theta)= 0 to oo
quadrant 4 (lower right) theta=270 to 360 degrees tan(theta)= -oo to 0

so tan(theta) = -1 has solutions in quadrant 2 and quadrant 4 with theta = 135 degrees and 315 degrees.