Convert the rectangular coordinates left parenthesis, 9, comma, minus, 3, square root of, 3, end square root, right parenthesis(9,−3 3 ) 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, 9, comma, minus, 3, square root of, 3, end square root, right parenthesis(9,−3 3 ) 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 (9, -3√3) into polar form, we need to find the radius (r) and the angle (θ).
Step 1: Calculate the radius (r) The formula to calculate the radius in polar coordinates is r = √(x² + y²). Here, x = 9 and y = -3√3.
r = √((9)² + (-3√3)²) r = √(81 + 27) r = √108 r = 6√3
Step 2: Calculate the angle (θ) The formula to calculate the angle in polar coordinates is θ = atan(y/x). Here, x = 9 and y = -3√3.
θ = atan((-3√3)/9) θ = -π/6
However, we want θ to be in the interval [0, 2π). So, we add 2π to the angle if it's negative.
θ = -π/6 + 2π θ = 11π/6
So, the polar coordinates are (6√3, 11π/6).
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