1. The problem is to find the value of \(\sin(2\theta)\) given a trigonometric context. 2. The double-angle formula for sine is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). 3. This formula is derived from the sum of angles identity: \(\sin(a+b) = \sin a \cos b + \cos a \sin b\), setting \(a = b = \theta\). 4. To solve a specific problem, you need values for \(\sin(\theta)\) and \(\cos(\theta)\). 5. For example, if \(\sin(\theta) = \frac{3}{5}\) and \(\cos(\theta) = \frac{4}{5}\), then: $$\sin(2\theta) = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}$$ 6. This means the sine of double the angle is \(\frac{24}{25}\). 7. Remember, the sine and cosine values must satisfy the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). 8. This method helps in solving many trigonometric problems involving double angles.