Reducing Radian Angle

Question

How would I reduce a large radian angle into a smaller one?

For example, if you're asked to find a related angle for $\cos \frac{3177 \pi}{12}.$ I usually try to break it down to $\frac{\pi}{2},$ $\frac{\pi}{3},$ $\frac{\pi}{4},$ or $\frac{\pi}{6}$ to visualize it in a unit circle, but I wouldn't know where the terminal arm would lie in such a big angle.

I know that a period for one revolution in a unit circle is equal to $2\pi.$ So I could divide the big angle $\frac{3177 \pi}{12}$ by $2\pi.$

So that's $\frac{3177 \pi}{12} \cdot \frac{1} {2 \pi}= 132.375.$ So I found out the number of rotations. But I still don't know how to interpret that in a unit circle. If I were to draw the angle $\frac{3177 \pi}{12},$ which quadrant would the terminal arm lie?

Answer 1

You are almost there. You have $$\frac{3177\pi/12}{2\pi}=132.375$$ So you can subtract 132 complete rotations $$3177\pi/12-132\cdot 2\pi=2\pi\frac{9}{24}=2\pi\frac{3}{8}$$ Less than $1/4$ of a circle would be in the first quadrant, $3/8$ is in the second quadrant. Second quadrant are angles between $1/4$ and $1/2$ of a full circle ($2\pi$)

Answer 2

The important thing you found is $.375$ because you have $132$ full rotations which are not important and the rest is $2\pi\cdot 0.375=\frac{3}{4}\pi$ so it is in the second quadrant.

Answer 3

You want to find the unique integer $k$ such that $$ 0\le\frac{3177\pi}{12}-2k\pi<2\pi $$ This inequality becomes $$ 0\le 3177-24k<24 $$ so you want the quotient of dividing $3177$ by $24$, but also the remainder is important.

\begin{array}{cccc|l} 3 & 1 & 7 & 7 & 24 \\ & 7 & 7 & & 132\\ & & 5 & 7 \\ & & & 9 \end{array} or $3177=132\cdot24+9$.

Hence $$ \frac{3177\pi}{12}=132\cdot2\pi+\frac{9\pi}{12}=132\cdot2\pi+\frac{3\pi}{4} $$