Compound And Multiple Angles

The controlling formula is the triple-angle identity $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$, which expresses the sine of a tripled angle entirely in terms of the sine of the original angle. This identity is built by combining the angle-addition formula with the double-angle formulas, so it lets a tripled angle be evaluated algebraically without ever finding the angle itself. Substituting the given value $\sin \theta =\frac{1}{2}$ directly, the expression becomes $3\cdot \frac{1}{2}-4\cdot {\left(\frac{1}{2}\right)}^3=\frac{3}{2}-4\cdot \frac{1}{8}=\frac{3}{2}-\frac{1}{2}=1$. The acute condition uniquely fixes $\theta =30^{\circ }$, so the tripled angle is $90^{\circ }$ and $\sin 90^{\circ }=1$, matching the algebraic result exactly. Option $\frac{\sqrt{3}}{2}$ confuses the answer with a cosine value. Option $\frac{1}{2}$ simply repeats the original sine without applying the tripling. Option $0$ misapplies the cubic term. This follows the standard triple-angle expansion pattern of JEE Advanced. As a final consistency check, the geometric route confirms that $\theta =30^{\circ }$ gives $3\theta =90^{\circ }$ and hence $\sin 3\theta =1$, in full agreement with the identity-based computation.