Question

Convert $(-√3,1)$ from rectangular coordinates to polar coordinates. (Use radians, with your angle in the interval $0,2π)$, and with $r > 0$.)

a) $r = 2, θ = 5π6$
b) $r = 2, θ = π6$
c) $r = 2, θ = 7π6$
d) $r = 2, θ = 11π6$

Answer 1

The rectangular coordinates (-√3,1) convert to polar coordinates as (r = 2, θ = 5π/6), falling in the second quadrant with radius 2 and angle 5π/6 radians.

Step-by-step explanation:

To convert the point (√3,1) from rectangular coordinates to polar coordinates, we need to find the magnitude of the vector (r) and its angle (θ) from the x-axis. To find the radius (r), we use the Pythagorean theorem:
r = √(x^2 + y^2) = √(√3^2 + 1^2) = √(3 + 1) = 2

The angle (θ) can be found using the arctangent function since the point lies in the second quadrant:
θ = tan^√1(y/x) = tan^√1(1/√3)
Since the point is in the second quadrant, we add π to the principal value to get θ in the correct range (0,2π). Thus, θ = π - tan^√1(1/√3).

The exact value of tan^√1(1/√3) is π/6, and adding π gives us:
θ = π - π/6 = 5π/6

Therefore, the polar coordinates are (r = 2, θ = 5π/6), which corresponds to option a).