Answer
a. $\displaystyle \overline{t}=\frac{\pi}{3}$
b. $\displaystyle \overline{t}=\frac{\pi}{3}$
c. $\displaystyle \overline{t}=\frac{\pi}{6}$
d. $\overline{t}=3.5-\pi\approx 0.36$
Work Step by Step
The reference number associated with the real number $t$ is the shortest distance along the unit circle between the terminal point determined by $t$ and the x-axis.
For each t, find each terminal point on the unit circle (positive=counterclockwise) and associate it with the terminal point of some t between 0 and $ 2\pi$
If the terminal point "lands" in quadrants II, III or IV,
choose the symmetric terminal number ($\pm\pi$) in quadrant I$:$
t in Q.II $\Rightarrow \overline{t}=\pi-t$
t in Q.III$\Rightarrow \overline{t}=t-\pi$
t in Q.IV$\Rightarrow \overline{t}=2\pi-t$
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a. The terminal point of $t=\displaystyle \frac{4\pi}{3}$ is in Q.III,
its reference number is $\displaystyle \frac{4\pi}{3}-\pi=\frac{\pi}{3}$
b. $ $The terminal point of $t=\displaystyle \frac{5\pi}{3}$ is in Q.IV,
its reference number is $2\pi- \displaystyle \frac{5\pi}{3}=\frac{\pi}{3}$
c. $\displaystyle \frac{-7\pi}{6}$ is in Q.II (clockwise from 0), and has the same terminal point as$ \displaystyle \frac{5\pi}{6}$
its reference number is $\displaystyle \frac{5\pi}{6}-\pi=\frac{\pi}{6}$
d. The terminal point of 3.5 is in Q.III, ($\pi\approx$3.14),
its reference number is $3.5-\pi\approx 3.5-3.14=0.36$
