Mathematics - Trigonometric Equation Question with Solution | TestHub

$$\begin{array}{rl}& 2\cos \theta \left[\cos 120^{\circ }+\cos 2\theta \right]=1\\ \Rightarrow & 2\cos \theta \left(-\frac{1}{2}+2{\cos }^2\theta -1\right)=1\\ \Rightarrow & 4{\cos }^3\theta -3\cos \theta -1=0\\ \Rightarrow & \cos 3\theta =1=\cos 0\\ \Rightarrow & 3\theta =2n\pi \text{ or }\theta =\frac{2n\pi }{3},n\in z\end{array}$$ Given the values so that $2n$ does not exceed 18 . Hence, the sum $=\frac{2\pi }{3}{\sum }_1^{9}n=\frac{2\pi }{3}\times \frac{9(9+1)}{2}=30\pi$