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

2 Answers
Dec 17, 2017

Quadrant 3

Explanation:

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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

Dec 17, 2017

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

Explanation:

Visualize a unit circle:

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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:

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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:

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And rotate our angle accordingly:

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It seems that our angle lands on Quadrant III.