Question

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

Answer 1

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.