Question

If the points (-2,0),(-1,1/root3) and (cos 4theta,sin 4theta) are collinear,then the number of values of theta when 0< equal to theta < equal to pi
Answer is 2 (explain)

Answer 1

$$\begin{gathered} \mathrm{Given}\mathrm{the}\mathrm{points}\left(-2,0\right),\left(-1,\frac{1}{\sqrt{3}}\right),\left(\cos 4\mathrm{\theta },\sin 4\mathrm{\theta }\right)\mathrm{are}\mathrm{collinear} \\ \mathrm{Therefore}, \\ \frac{1}{2}\left[-2\left(\frac{1}{\sqrt{3}}-\sin 4\mathrm{\theta }\right)-1\left(\sin 4\mathrm{\theta }-0\right)+\cos 4\mathrm{\theta }\left(0-\frac{1}{\sqrt{3}}\right)\right]=0 \\ \Rightarrow \frac{-2}{\sqrt{3}}+2\sin 4\mathrm{\theta }-\sin 4\mathrm{\theta }-\frac{1}{\sqrt{3}}\cos 4\mathrm{\theta }=0 \\ \Rightarrow \sqrt{3}\sin 4\mathrm{\theta }-\cos 4\mathrm{\theta }=2 \\ \Rightarrow \frac{\sqrt{3}}{2}\sin 4\mathrm{\theta }-\frac{1}{2}\cos 4\mathrm{\theta }=1 \\ \Rightarrow \sin \left(4\mathrm{\theta }-\frac{\mathrm{\pi }}{6}\right)=1 \\ \Rightarrow \sin \left(4\mathrm{\theta }-\frac{\mathrm{\pi }}{6}\right)=\sin \frac{\mathrm{\pi }}{2} \\ \Rightarrow 4\mathrm{\theta }-\frac{\mathrm{\pi }}{6}=\frac{\mathrm{\pi }}{2} \\ \Rightarrow 4\mathrm{\theta }=\frac{4\mathrm{\pi }}{6} \\ \Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{6} \\ \mathrm{or}, \\ \Rightarrow \sin \left(4\mathrm{\theta }-\frac{\mathrm{\pi }}{6}\right)=1 \\ \Rightarrow \sin \left(4\mathrm{\theta }-\frac{\mathrm{\pi }}{6}\right)=\sin \frac{8\mathrm{\pi }}{3} \\ \Rightarrow 4\mathrm{\theta }-\frac{\mathrm{\pi }}{6}=\frac{8\mathrm{\pi }}{3} \\ \Rightarrow 4\mathrm{\theta }=\frac{8\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{6} \\ \Rightarrow \mathrm{\theta }=\frac{17\mathrm{\pi }}{24} \end{gathered}$$