Answer
a. $\displaystyle \overline{t}=\frac{2\pi}{9}$
b. $\displaystyle \overline{t}=\frac{2\pi}{9}$
c. $\overline{t}\approx 0.14$
d. $\overline{t}\approx 1.28$
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 $\displaystyle \frac{7\pi}{9}$ is in Q.II ($\pi= \displaystyle \frac{9\pi}{9}$),
its reference number is $\pi- \displaystyle \frac{7\pi}{9}=\frac{2\pi}{9} $
b. The terminal point of $-\displaystyle \frac{7\pi}{9}$ is in Q.III (clockwise, $-\pi=- \displaystyle \frac{9\pi}{9}$),
the same as the terminal point of $\displaystyle \frac{11\pi}{9}$
its reference number is $\displaystyle \frac{11\pi}{9}-\pi=\frac{2\pi}{9} $
c. The terminal point of $t=-3$ is in Q.III (clockwise, $-\pi\approx-3.14)$
is the same terminal point as $\approx\pi+0.14$ ,
its reference number is $\approx\pi+0.14-\pi=0.14$
d. The terminal point of 5 is in Q.III, (2$\pi\approx$6.28),
its reference number is $2\pi-5\approx 1.28$