Question
Convert $\theta = -\frac{2a\pi }{3}$ rad to degrees form where $a$ is a natural number. Hence, find
the smallest value of $𝑎$ if $𝜃$ lies in the second quadrant.
I trying to solve …
$$\theta = -\frac{2a\pi }{3}$$
I know $\pi$ rad $=180^{\circ}$
so $1$ rad $=\frac{180}{\pi}$
then
$$\theta = -\frac{2a\pi }{3}\times\frac{180}{\pi}=-120a^{\circ} $$
𝜃 lies in the second quadrant then $𝜃$ lies in the second quadrant equal to $180^{\circ}-\theta$ ???
I dont know how to start to get $𝜃$ in second quadrant. May you help me to solve this question?
Thank you so much.
Best Answer
You get the same position in the unit circle if you add multiples of $360^\circ$. Second quadrant means angles between $90^\circ$ and $180^\circ$. So take a look at several $a$ values: $a=0$ means $\theta=0$, $a=1$ means $\theta=-120\to -120+360=240$. None of these are in the second quadrant. For $a=2$, $\theta=-240\to -240+360=120$. This is the right answer