∣ cos 3 θ − 8 − 12 cos 2 θ 3 3 1 1 3 ∣ = 0 \left|\begin{array}{ccc}\cos 3 \theta & -8 & -12\\ \cos 2 \theta & 3 & 3\\ 1 & 1 & 3\end{array}\right|=0 cos 3 θ cos 2 θ 1 − 8 3 1 − 12 3 3 = 0 ∣ cos 3 θ − 8 − 4 cos 2 θ 3 1 1 1 1 ∣ = 0 \left|\begin{array}{ccc}\cos 3 \theta & -8 & -4\\ \cos 2 \theta & 3 & 1\\ 1 & 1 & 1\end{array}\right|=0 cos 3 θ cos 2 θ 1 − 8 3 1 − 4 1 1 = 0 ( C 1 → C 1 − C 2 ) & C 2 → C 2 − C 3 \left(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}\right) \& \mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{3} ( C 1 → C 1 − C 2 ) & C 2 → C 2 − C 3 ∣ cos 3 θ + 8 − 4 − 3 cos 2 θ − 3 2 1 0 0 1 ∣ = 0 \left|\begin{array}{ccc}\cos 3 \theta+8 & -4 & -3 \\ \cos 2 \theta-3 & 2 & 1\\ 0 & 0 & 1\end{array}\right|=0 cos 3 θ + 8 cos 2 θ − 3 0 − 4 2 0 − 3 1 1 = 0 2 cos 3 θ + 16 + 4 cos 2 θ − 12 = 0 2 \cos 3 \theta+16+4 \cos 2 \theta-12=0 2 cos 3 θ + 16 + 4 cos 2 θ − 12 = 0 ( 4 cos 3 θ − 3 cos θ ) + 2 ( 2 cos 2 θ − 1 ) + 2 = 0 \left(4 \cos ^{3} \theta-3 \cos \theta\right)+2\left(2 \cos ^{2} \theta-1\right)+2=0 ( 4 cos 3 θ − 3 cos θ ) + 2 ( 2 cos 2 θ − 1 ) + 2 = 0 4 cos 3 θ + 4 cos 2 θ − 3 cos θ = 0 4 \cos ^{3} \theta+4 \cos ^{2} \theta-3 \cos \theta=0 4 cos 3 θ + 4 cos 2 θ − 3 cos θ = 0 cos θ ( 4 cos 2 θ + 4 cos θ − 3 ) = 0 \cos \theta\left(4 \cos ^{2} \theta+4 \cos \theta-3\right)=0 cos θ ( 4 cos 2 θ + 4 cos θ − 3 ) = 0 cos θ ( 2 cos θ + 3 ) ( 2 cos θ − 1 ) = 0 \cos \theta(2 \cos \theta+3)(2 \cos \theta-1)=0 cos θ ( 2 cos θ + 3 ) ( 2 cos θ − 1 ) = 0 cos θ = 0 , 1 2 , − 3 2 \cos \theta=0, \frac{1}{2},-\frac{3}{2} cos θ = 0 , 2 1 , − 2 3 θ = π 2 , 3 π 2 , π 3 , 5 π 3 \theta=\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{\pi}{3}, \frac{5 \pi}{3} θ = 2 π , 2 3 π , 3 π , 3 5 π Sum = 4 π \operatorname{Sum}=4 \pi Sum = 4 π