Convert the rectangular coordinates left parenthesis, 6, comma, minus, 6, right parenthesis(6,−6) 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, 6, comma, minus, 6, right parenthesis(6,−6) 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, -6) into polar form, we follow these steps:
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Calculate the radius r. The formula for r is sqrt(x^2 + y^2). Substituting x = 6 and y = -6, we get r = sqrt((6)^2 + (-6)^2) = sqrt(36 + 36) = sqrt(72) = 6sqrt(2).
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Calculate the angle θ. The formula for θ is arctan(y/x). Substituting y = -6 and x = 6, we get θ = arctan((-6)/6) = arctan(-1). However, because we're in the fourth quadrant (where x is positive and y is negative), we need to add π to the result to get the angle in the correct quadrant. So, θ = arctan(-1) + π = π - π/4 = 7π/4.
So, the polar coordinates are (6sqrt(2), 7π/4).
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