The least positive angle measured in degrees satisfying the equation  ${\mathbf{sin}}^3\mathbf{x}\mathbf{+}{\mathbf{sin}}^3\mathbf{2}\mathbf{x}\mathbf{+}{\mathbf{sin}}^3\mathbf{3}\mathbf{x}\mathbf{=}{\left(\mathbf{sinx}\mathbf{+}\mathbf{sin}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{sin}\mathbf{3}\mathbf{x}\right)}^3$

${\text{sin}}^{\text{3}}{\text{x+\hspace{0.17em}sin}}^{\text{3}}{\text{\hspace{0.17em}2x+sin}}^{\text{3}}\text{3x=}{\left(\text{sinx\hspace{0.17em}+\hspace{0.17em}sin\hspace{0.17em}2x\hspace{0.17em}+\hspace{0.17em}sin\hspace{0.17em}3x}\right)}^{\text{3}}$

$\left(\therefore {\left(\text{a\hspace{0.17em}+\hspace{0.17em}b\hspace{0.17em}+\hspace{0.17em}c}\right)}^{\text{3}}{\text{=a}}^{\text{3}}{\text{\hspace{0.17em}+\hspace{0.17em}b}}^{\text{3}}{\text{\hspace{0.17em}+\hspace{0.17em}c}}^{\text{3}}\right)\Rightarrow \left(\text{a+b}\right)\left(\text{b+c}\right)\left(\text{c+a}\right)\text{\hspace{0.17em}=0}$

$\Rightarrow \left(\text{sin\hspace{0.17em}x\hspace{0.17em}+\hspace{0.17em}sin\hspace{0.17em}2x}\right)\left(\text{sin2x\hspace{0.17em}+\hspace{0.17em}sin\hspace{0.17em}3x}\right)\left(\text{sin\hspace{0.17em}3x\hspace{0.17em}+\hspace{0.17em}sinx}\right)\text{\hspace{0.17em}=\hspace{0.17em}0}$

$\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}2sin}\frac{\text{3x}}{\text{2}}\text{cos}\frac{\text{x}}{\text{2}}\text{=0\hspace{0.17em}}\left(\text{or}\right)\text{2sin}\frac{\text{5x}}{\text{2}}\text{cos}\frac{\text{x}}{\text{2}}\text{=0}$

(or) 2 sin 2x cos x = 0

$\therefore$ Least positive angle is $\frac{\text{2π}}{\text{5}}{\text{=72}}^{\text{0}}$