Question

If $\sin x+\sin 2x+\sin 3x=2,$ then the value of $x$ is

• A. ${30}^0$
• B. ${45}^0$
• C. ${60}^0$
• D. None of these

Answer 1

Answer: (D)

None of these

Given $\sin x+\sin 2x+\sin 3x=2$

Let us examine if the above statement is true for the given options-

$\theta ={30}^{\circ }\to \sin {30}^{\circ }+\sin {60}^{\circ }+\sin {90}^{\circ }=\frac{1}{2}+\frac{\sqrt{3}}{2}+1\ne 2\theta ={45}^{\circ }\to \sin {45}^{\circ }+\sin {90}^{\circ }+\sin {135}^{\circ }=\frac{1}{\sqrt{2}}+1+\frac{1}{\sqrt{2}}\ne 2[\because \sin {135}^{\circ }=\sin (90^{\circ }+{45}^{\circ })=\cos {45}^{\circ }=\frac{1}{\sqrt{2}}]\theta ={60}^{\circ }\to \sin {60}^{\circ }+\sin {120}^{\circ }+\sin {180}^{\circ }=\frac{\sqrt{3}}{2}+\cos {30}^{\circ }+\cos {90}^{\circ }$

$=\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}+0\ne 2$