Let 0<θ1<θ2<θ3<… denote the positive solution of the equation 3+3cos⁡θ=2sin2⁡θ.  The value of θ3+θ7 is

3+3cos⁡θ=2−2cos2⁡θ⇒ 2cos2⁡θ+3cos⁡θ+1=0⇒ (2cos⁡θ+1)(cos⁡θ+1)=0∴ cos⁡θ=−1 or cos⁡θ=−12If  cos⁡θ=−1 then θ=π,3π,5π,7π,9π,…If cos⁡θ=−1/2 then θ=2π3,4π3,8π3,10π3,14π3,…Solutions in increasing order are     2π3<π<4π3<8π3<3π<10π3<14π3<5π,…∴ θ3+θ7=4π3+14π3=6π