Question

Given point Q equals negative 6 radical 3 comma negative 6 in rectangular coordinates, what are the corresponding polar coordinates?

Answer 1

The corresponding polar coordinates for the point Q(-6√3, -6) are: Option D: (12, 7π/6).

How to find the Polar Coordinates?

To convert the point Q(-6√3, -6) from rectangular coordinates to polar coordinates, we need to find the radial distance (r) and the angle (θ) that Q makes with the positive x-axis.

The radial distance (r) can be calculated using the formula:

r = √(x² + y²)

Given x = -6√3 and y = -6:

Thus:

r = √((-6√3)² + (-6)²)

r = √(108 + 36)

r = √144

r = 12

Now, to find the angle (θ), we can use the formula:

θ = tan⁻¹(y / x)

θ = tan⁻¹((-6) / (-6√3))

θ = tan⁻¹(1/√3)

θ = π/6

Since the point Q is in the third quadrant (both x and y are negative), the angle θ needs to be adjusted accordingly.

In the third quadrant, θ = π + π/6 = 7π/6.

So, the corresponding polar coordinates for the point Q(-6√3, -6) are (r, θ) = (12, 7π/6).

Answer 2

Given the rectangular coordinates of point Q:

`Q(-6√(3),-6)`

You need to remember that the form from rectangular oordinates to polar coordinates is:

`(x,y)\rightarrow(r,\theta)`

By definition:

`\begin{gathered} r=√(x^2+y^2) \\ \\ \theta=tan^(-1)((y)/(x)) \end{gathered}`

In this case, you can identify that:

`\begin{gathered} x=-6√(3) \\ y=-6 \end{gathered}`

Then, you can determine that:

`\begin{gathered} r=\sqrt{(-6√(3))^2+(-6)^2}=12 \\ \\ \theta=tan^(-1)((-6)/(-6√(3)))=(5\pi)/(6) \end{gathered}`

Therefore, the polar coordinates are:

`(12,(5\pi)/(6))`

Hence, the answer is: Second option.