How do you convert (-2, -2sqrt{3})(2,23)into polar coordinates?

1 Answer
May 5, 2018

(4, (4pi)/3)(4,4π3) (radians) or (4, 240^@)(4,240) (degrees)

Explanation:

Rectangular -> Polar: (x, y) -> (r, theta)(x,y)(r,θ)

  • Find rr (radius) using r = sqrt(x^2 + y^2)r=x2+y2
  • Find thetaθ by finding the reference angle: tantheta = y/xtanθ=yx and use this to find the angle in the correct quadrant

r = sqrt((-2)^2 + (-2sqrt3)^2)r=(2)2+(23)2

r = sqrt(4+(4*3))r=4+(43)

r = sqrt(4+12)r=4+12

r = sqrt(16)r=16

r = 4r=4

Now we find the value of thetaθ using tantheta = y/xtanθ=yx.

tantheta = (-2sqrt3)/-2tanθ=232

tantheta = sqrt3tanθ=3

theta = tan^-1(sqrt3)θ=tan1(3)

theta = (pi)/3θ=π3 or (4pi)/34π3

To determine which one it is, we have to look at our coordinate (-2, -2sqrt3)(2,23). First, let's graph it:
enter image source here

As you can see, it is in the third quadrant. Our thetaθ has to match that quadrant, meaning that theta = (4pi)/3θ=4π3.

From rr and thetaθ, we can write our polar coordinate:
(4, (4pi)/3)(4,4π3) (radians) or (4, 240^@)(4,240) (degrees)

Hope this helps!