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

Mar 8, 2018

Quadrants are broken up into regions of $\frac{\pi}{2}$, so if you calculate the smallest multiple of $\frac{\pi}{2}$ 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 $2 \pi$ radians, then the quadrants would be 1/4 of that angle... or $\frac{\pi}{2}$ for each quadrant.

q1 would range from 0 to $\frac{\pi}{2}$
q2 would range from $\frac{\pi}{2}$ to $\pi$
q3 would range from $\pi$ to $\frac{3 \pi}{2}$
q4 would range from $\frac{3 \pi}{2}$ to $2 \pi$

to make these comparable to the desired angle, both the value of $\frac{7 \pi}{5}$ and the quadrant values need to be brought to their lowest common denominator. In this case, that is 10.

angle: $\frac{14 \pi}{10}$

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: $\frac{10 \pi}{10} \leftrightarrow \frac{15 \pi}{10}$
q4: $\frac{15 \pi}{10} \leftrightarrow \frac{20 \pi}{10}$

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