How do you determine the quadrant in which #(7pi)/5# lies?

1 Answer
Ben
Mar 8, 2018

Quadrants are broken up into regions of #pi/2#, so if you calculate the smallest multiple of #pi/2# that is larger than your angle, you can determine the quadrant (3rd quadrant for this one)

Explanation:

I'm referring to quadrants as q1, q2, q3, and q4 for brevity:

if traveling around a unit circle is #2pi# radians, then the quadrants would be 1/4 of that angle... or #pi/2# for each quadrant.

q1 would range from 0 to #pi/2#
q2 would range from #pi/2# to #pi#
q3 would range from #pi# to #(3pi)/2#
q4 would range from #(3pi)/2# to #2pi#

to make these comparable to the desired angle, both the value of #(7pi)/5# and the quadrant values need to be brought to their lowest common denominator. In this case, that is 10.

angle: #(14pi)/10#

we know that 14/10 is greater than 1, so we can eliminate q1 and q2 from the possible answers. This leaves us with q3 and q4:

q3: #(10pi)/10 harr (15pi)/10#
q4: #(15pi)/10 harr (20pi)/10#

Since 14 is less than 15, we can conclude that it is the third quadrant.

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