Convert the rectangular coordinates left parenthesis, minus, 3, square root of, 3, end square root, comma, minus, 9, right parenthesis(−3 3 ,−9) into polar form. Express the angle using radians in terms of piπ over the interval 0, is less than or equal to, theta, is less than, 2, pi0≤θ<2π, with a positive value of rr.
Question
Convert the rectangular coordinates left parenthesis, minus, 3, square root of, 3, end square root, comma, minus, 9, right parenthesis(−3 3 ,−9) into polar form. Express the angle using radians in terms of piπ over the interval 0, is less than or equal to, theta, is less than, 2, pi0≤θ<2π, with a positive value of rr.
Solution
To convert rectangular coordinates to polar form, we use the following formulas:
r = sqrt(x^2 + y^2) θ = atan(y/x)
Given the rectangular coordinates (-3√3, -9), we can substitute these into the formulas:
r = sqrt((-3√3)^2 + (-9)^2) r = sqrt(27 + 81) r = sqrt(108) r = 2√27
Next, we calculate θ:
θ = atan(-9 / -3√3) θ = atan(3 / √3) θ = atan(√3) θ = π/3
However, because the point is in the third quadrant, we need to add π to the angle to get the correct polar angle. So, θ = π/3 + π = 4π/3.
So, the polar coordinates are (2√27, 4π/3).
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