Explanation:

- Given: $\sin 2\theta =\cos 3\theta$ and ? is an acute angle.

- Using the identity, $\cos 3\theta =\sin ({90}^{\circ }-3\theta )$, we equate $\sin 2\theta =\sin ({90}^{\circ }-3\theta )$.

- Since ? is acute, set $2\theta ={90}^{\circ }-3\theta$.

- Solving: $2\theta +3\theta ={90}^{\circ }\Rightarrow 5\theta ={90}^{\circ }\Rightarrow \theta ={18}^{\circ }$.

- With $\theta ={18}^{\circ }$, $\sin {18}^{\circ }=\frac{\sqrt{5}-1}{4}$.

- Therefore, $1+4\sin \theta =1+4\times \frac{\sqrt{5}-1}{4}=1+(\sqrt{5}-1)=\sqrt{5}$.

- The correct answer is Option:3 v5

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