Question

The Cartesian coordinates of a point are given.(a) (6, −6)(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.(r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.(r, θ) = (b) (−1, 3)(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.(r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.(r, θ) =

Answer 1

To find the polar coordinates (r, θ) of a point given in Cartesian coordinates (x, y), we can use the equations (r, θ) = (√(x² + y²), atan2(y, x)).

Step-by-step explanation:

To find the polar coordinates (r, θ) of a point given in Cartesian coordinates (x, y), we can use the following equations:

(r, θ) = (√(x² + y²), atan2(y, x))

(a)(i) For the Cartesian coordinates (6, -6),

(r, θ) = (√(6² + (-6)²), atan2(-6, 6))

(r, θ) = (√(36 + 36), atan2(-6, 6))

(r, θ) = (6√2, -45°)

(a)(ii) For the Cartesian coordinates (6, -6),

(r, θ) = (-√(6² + (-6)²), atan2(-6, 6) + π)

(r, θ) = (-6√2, -45° + π)

(b)(i) For the Cartesian coordinates (-1, 3),

(r, θ) = (√((-1)² + 3²), atan2(3, -1))

(r, θ) = (√(1 + 9), atan2(3, -1))

(r, θ) = (√10, 116.57°)

(b)(ii) For the Cartesian coordinates (-1, 3),

(r, θ) = (-√((-1)² + 3²), atan2(3, -1) + π)

(r, θ) = (-√10, 116.57° + π)