Convert the rectangular coordinates left parenthesis, minus, 6, square root of, 3, end square root, comma, 0, right parenthesis(−6 3 ,0) 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, 6, square root of, 3, end square root, comma, 0, right parenthesis(−6 3 ,0) 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 the rectangular coordinates (-6√3, 0) into polar form, we need to find the radius (r) and the angle (θ).
Step 1: Calculate the radius (r) The radius in polar coordinates is given by the formula r = √(x² + y²). Here, x = -6√3 and y = 0. So, r = √((-6√3)² + 0²) = √(108) = 6√3.
Step 2: Calculate the angle (θ) The angle in polar coordinates is given by the formula θ = atan(y/x). Here, y = 0 and x = -6√3. So, θ = atan(0/(-6√3)) = 0. However, since x is negative, we are in the second or third quadrant. But since y is also 0, we are on the negative x-axis, which is equivalent to an angle of π.
So, the polar coordinates are (6√3, π).
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