Question

How do you determine the quadrant in which `(9pi)/8` lies?

Answer 1

Quadrant 3

Explanation:

We can see from the drawring that `(9pi)/8` is just over `pi`
As its `1.125 * pi` So hence in the 3rd quadrant

As `pi < (9pi)/8 < (3pi)/2`

Answer 2

I would visualize its rotation in the unit circle to find out that it is in Quadrant III.

Explanation:

Visualize a unit circle:

The angle `9/8 pi` is a counterclockwise rotation of `9/8 pi` radians from the starting point, which is in case of the unit circle, `(1, 0)`.

Since `9/8 pi` is `8/8 pi + 1/8 pi`, simplify out `8/8` to get `pi + 1/8 pi`.

This tells me, intuitively, to rotate `pi` radians, and then another `1/8` of `pi`. We first rotate counterclockwise by `pi` radians, which is half of a full rotation:

Then, since we have `pi` radians left to complete a full rotation, yet our angle only tells us to rotate `1/8 pi` more, let's slice up the remaining semicircle into `8` parts:

And rotate our angle accordingly:

It seems that our angle lands on Quadrant III.