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

1 Answer
Shantelle
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+(−2√3)2

r = sqrt(4+(4*3))r=√4+(4⋅3)

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=−2√3−2

tantheta = sqrt3tanθ=√3

theta = tan^-1(sqrt3)θ=tan−1(√3)

theta = (pi)/3θ=π3 or (4pi)/34π3

To determine which one it is, we have to look at our coordinate (-2, -2sqrt3)(−2,−2√3). 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!

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