Find the polar coordinates

Question

Write the point (x, y) = (−1,√3) in polar coordinates (r, θ)

Answer 1

What do we know? r=(x²⁺y²)^1/2 and tanθ=y/x x=-1 and y=√3 just substitute r=(4)^1/2=2

tanθ=√3/-1 θ=tan^-1(√3/-1), (2,-60)= (-2,-60+180)=(-2,120) since the xy coordinates are in the second quadrant. (r,θ)=(-r,θ+180), reference Calculus Larson and B. Edwards 7e edition p719

Answer 2

If we plot the point whose coordinates are given as ( -1, √3 ), we end up in the 2nd quadrant with a special right triangle ( a 30-60-90 ) whose hypotenuse is 2 ( via Pythagorean Theorem or the special rules for 3-60-90 right triangles). The reference angle for this triangle is 60 degrees, so the original angle of rotation from the positive x-axis would be 120°. Expressing this in polar coordinates, our "r" value is 2 and the value for θ, in radians, is 2π/3, so the coordinates are ( 2, 2π/3).