Question

# A real number t is given. Find the reference number for t. $t=\frac{5 \pi}{3}$

Solution

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The following formulas are important for finding the reference number $\bar{t}$:

\begin{align*} &\text{Quadrant I:} &&\bar{t}=t \\ &\text{Quadrant II:} &&\bar{t}=\pi-t \\ &\text{Quadrant III:} &&\bar{t}=t-\pi \\ &\text{Quadrant IV:} &&\bar{t}=2\pi-t \\ \end{align*}

Our angle $t$ is in the fourth quadrant because $\dfrac {3\pi} 2 \leq t \leq 2\pi$. Hence, the reference number for $t$ is:

$\bar{t}=2\pi-\dfrac{5\pi}{3}=\dfrac{\pi}{3}$

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