How do you estimate an angle to the nearest one-half radian?

Start with drawing of trigonometric circle (or you can find one in book or online).
Now that you have trigonometric circle in front of you, locate four main rays representing angles ${\displaystyle 0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}}$.
Also, locate rays that represent angles ${\displaystyle \frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}}$.
If you located these rays, it is now simple to approximate an angle to the nearest ${\displaystyle \frac{\pi }{2}}$. For example, if the angle is between rays that represent ${\displaystyle \frac{\pi }{4}}$ and ${\displaystyle \frac{\pi }{2}}$ we round that andle to ${\displaystyle \frac{\pi }{2}}$.