Angular Displacement

• A wheel turns through 5 radians. This is pretty straightforward: $\mathrm{\Delta }\theta =5\mathrm{rad}$
• A wheel spins through 90°. To convert from degrees to radians, you can multiply by $\frac{\pi \mathrm{rad}}{{180}^{\circ }}$: $$\mathrm{\Delta }\theta ={90}^{\circ }\times \frac{\pi \mathrm{rad}}{{180}^{\circ }}=\frac{\pi }{2}\mathrm{rad}$$
• A wheel spins around 5 times. "Times" is usually a synonym for "revolutions". To convert from revolutions to radians, multiply by $\frac{2\pi \mathrm{rad}}{1\mathrm{rev}}$: $$\mathrm{\Delta }\theta =5\mathrm{rev}\times \frac{2\pi \mathrm{rad}}{1\mathrm{rev}}=10\pi \mathrm{rad}$$

Remember that a counterclockwise rotation means $\mathrm{\Delta }\theta >0$, and a clockwise rotation means $\mathrm{\Delta }\theta <0$.