Evaluate $\sin \left(2{\sin }^{-1}\frac{7}{25}\right)$.

Step-by-step solution

1. Let $\theta ={\sin }^{-1}\frac{7}{25}$, then $\sin \theta =\frac{7}{25}$ and $\theta \in [-\frac{\pi }{2},\frac{\pi }{2}]$.
2. Find $\cos \theta =\sqrt{1-{\sin }^2\theta }=\sqrt{1-\frac{49}{625}}=\frac{24}{25}$ (positive in the principal range).
3. Use the double‑angle identity $\sin 2\theta =2\sin \theta \cos \theta$.
4. Substitute the values: $\sin 2\theta =2\cdot \frac{7}{25}\cdot \frac{24}{25}=\frac{336}{625}$.
5. Therefore $\sin \left(2{\sin }^{-1}\frac{7}{25}\right)=\frac{336}{625}$.

Concept tested: Double‑angle using inverse sine

Published 16 September 2026 · How this solution was prepared and checked