Answer
a. $\displaystyle \overline{t}=\frac{\pi}{5}$
b. $\displaystyle \overline{t}=\frac{2\pi}{7}$
c. $\overline{t}\approx 0.28$
d. $\overline{t}\approx 0.72$
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 $\displaystyle \frac{11\pi}{5}=2\pi+\frac{\pi}{5}$ is in Q.I ,
the same as the terminl point for $\displaystyle \frac{\pi}{5}$
its reference number is $\displaystyle \frac{\pi}{5} $.
b. The terminal point of $-\displaystyle \frac{9\pi}{7}=-\pi-\frac{2\pi}{7}$ is in Q.II (clockwise, $-\pi=- \displaystyle \frac{9\pi}{9}$),
the same as the terminal point of $\displaystyle \frac{5\pi}{7}$
its reference number is $\displaystyle \pi-\frac{5\pi}{7}=\frac{2\pi}{7} $
c. The terminal point of $6$ is in Q.III, (2$\pi\approx$6.28),
its reference number is $2\pi-6\approx 0.28$
d. The terminal point of $t=7$ is in Q.I (2$\pi\approx$6.28)
is the same terminal point as $\approx\pi+0.72$ ,
its reference number is $\approx 0.72$
