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:
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!