Question

Convert the rectangular coordinates (5,−5√3) to polar form. Let r>0 and 0≤θ<2π.

Enter your answer by filling in the boxes. Enter coordinates as simplifed fractions or radicals in simplest form.

( , )

Answer 1

(10, 5π/3)

Explanation:

To convert rectangular coordinates (5, -5√3) to polar form, we can use the following formulas:

r = √(x^2 + y^2)

θ = arctan(y/x)

Substituting the given values, we get:

r = √(5^2 + (-5√3)^2) = √(25 + 75) = √100 = 10

θ = arctan((-5√3)/5) = arctan(-√3) = -π/3

Note that the value of θ is in the fourth quadrant, which corresponds to a negative angle. However, we need to express the angle θ in the range 0 ≤ θ < 2π. To do this, we can add 2π to the angle if it is negative:

θ = -π/3 + 2π = (5π/3)

Therefore, the rectangular coordinates (5, -5√3) in polar form are (10, 5π/3).