The equation of wave form is given by

$v=Vm \sin \theta_1 \\ v=0.866V_m \theta_1=? \\ 0.866V_m=V_m \sin \theta_1 \\ \theta_1=\dfrac{\pi}{3} \\ Internal \\ 0 \lt \theta \lt \dfrac{\pi}{3} \hspace{2cm} v=V_m \sin \theta \\ \dfrac{\pi}{3} \lt 0 \lt \dfrac{\pi}{2} \hspace{1.8cm} v=0.866 V_m \\ \dfrac{\pi}{3} \lt \theta \lt \pi \hspace{2cm} v=V_m \sin \theta \\ V_{avg}=\int\limits^{\pi}_0\dfrac{vd \theta}{\pi} \\ =\dfrac{1}{\pi}\bigg\{\int\limits^{\pi\beta}_0vd\theta+\int\limits^{\pi/2}_{\pi \beta}vd\theta+\int\limits^{\pi}_{\pi/2}vd\theta\bigg\} \\ =\dfrac{1}{\pi}\bigg\{V_m\int\limits^{\pi\beta}_0\sin vd\theta+0.866V_m\int\limits^{\pi/2}_{\pi \beta}d\theta+V_m\int\limits^{\pi}_{\pi/2}\sin \theta d\theta\bigg\} \\ =\dfrac{V_m}{\pi}\{[-\cos \theta]^{\pi\beta}_0+0.866[\theta]^{\pi/2}_{\pi\beta}+[-\cos]^{\pi}_{\pi/2}\} \\ =\dfrac{V_m}{\pi}\bigg\{[-0.5-(-1)]+0.866\bigg[\dfrac\pi2-\dfrac\pi2\bigg]+[(-1)-(-0)]\bigg\} \\ =\dfrac{V_m}{\pi}\bigg\{-0.5+10.866\times\dfrac\pi6+1\bigg\} \\ =\dfrac{V_m}{\pi}\times\{1.5+0.4534\} \\ =0.6218 V_m$