Given any Cartesian coordinates, (x,y), there are polar coodinates

Question

Given any Cartesian coordinates, (x,y), there are polar coordinates (r,θ) with (-pie/2) < θ ≤ (pie/2).

Find polar coordinates with (-pie/2) < θ ≤ (pie/2) for the following Cartesian coordinates:

 

a) If (x,y)= (2,-3) then (r,θ)= ( _________ , __________ )

b) If (x,y)= (2,-2) then (r,θ)= ( _________, __________)

c) If (x,y)= (-9,-9) then (r,θ)= ( _________, __________)

d) If (x,y)= (1,-3) then (r,θ)= ( __________, __________)

e) If (x,y)= (-8,4) then (r,θ)= ( __________, __________)

f) If (x,y)= (0,7) then (r,θ)= ( _________, ___________)

Answer 1

Since the directions indicate you must use theta between -pi/2 and pi/2 when (x,y) is in the 2nd or 3rd quadrant you will have to use the negative square root of X² + y².

 

For part a) r = sqrt(4+9) = sqrt(13).

 

Theta = tan^-1(y/x) = tan^-1(-3/2) = -0.983 radians. (your calculator will give you inverse tangent between -pi/2 and pi/2)

 

Since (2,-3) is in quadrant 4, use the positive square root for r to set the polar coordinates:

 

sqrt(13), -0.983  (r, theta)

Answer 2

Use Pythagoras to determine r.

Then you can use x, y, and r to determine the angle.