Which of the following are polar coordinates of the point whose rectangular coordinates are (-2, -2sqrt3)?
a.) (4, pi/3)
b.) (2sqrt3, 4pi/3)
c.) (4, 7pi/6)
d.) (4, 4pi/3)
e.) none of these
recall that
r^2 = x^2 + y^2
tanθ = y/x
So see which coordinates (if any) fit those values.
To convert rectangular coordinates to polar coordinates, we can use the following formulas:
r = sqrt(x^2 + y^2)
θ = arctan(y / x)
Given the rectangular coordinates (-2, -2sqrt3), we can substitute these values into the formulas:
r = sqrt((-2)^2 + (-2sqrt3)^2)
r = sqrt(4 + 12)
r = sqrt(16)
r = 4
θ = arctan((-2sqrt3) / (-2))
θ = arctan(sqrt3)
θ ≈ π/3
Now that we have found the values for r and θ, we can check the multiple-choice answers:
a.) (4, π/3) - This matches our calculated values.
b.) (2sqrt3, 4π/3) - The value for r is incorrect, so this is not the correct answer.
c.) (4, 7π/6) - The value for θ is not equal to π/3, so this is not the correct answer.
d.) (4, 4π/3) - The value for θ is not equal to π/3, so this is not the correct answer.
e.) none of these - Since option a matches our calculated values, the correct answer is option a.
Therefore, the polar coordinates of the point whose rectangular coordinates are (-2, -2sqrt3) are (4, π/3), which is option a.