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.