$- \frac { 5 \pi } { 3 }$ is located between which of the following two numbers on the number line?
A) $- 5 \text { and } - 4$
B) $- 6 \text { and } - 5$
C) $- 7 \text { and } - 6$
D) $- 8 \text { and } - 7$

Question

Quizplus · Accepted Answer

To determine where $- \frac {5 \pi }{3}$ is located on the number line, we need to first find a reference angle between 0 and $2\pi$ (one full revolution) that is coterminal with $-\frac{5\pi}{3}$.

$-\frac{5\pi}{3}$ is $+\frac{\pi}{3}$ radians short of a full revolution in the clockwise direction from the 0 radians reference point.

Therefore, a coterminal angle in standard position in the counterclockwise direction would be:

\begin{align*}
2\pi - \frac{\pi}{3} &= \frac{6}{3}\pi - \frac{1}{3}\pi\
&= \frac{5}{3}\pi
\end{align*}

Since $\frac{5}{3}\pi$ is $\frac{2}{3}\pi$ radians away from the reference point of $0$, we can see that $- \frac {5 \pi }{3}$ is located on the number line between $-6$ and $-5$, so the answer is B.

−5π3- \frac { 5 \pi } { 3 }−35π​ Is Located Between Which of the Following Two Numbers on Located

$- \frac { 5 \pi } { 3 }$ Is Located Between Which of the Following Two Numbers on Located