Verified Answer :

  • Start with the double-angle identity: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).
  • Use the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to find \(\cos(\theta) = \sqrt{1 - (\frac{3}{5})^2} = \frac{4}{5}\).
  • Substitute the known values into the identity: \(\sin(2\theta) = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}\).
  • State the final result: \(\sin(2\theta) = \boxed{\frac{24}{25}}\).

Explanation

  1. Analysis of the Problem and Required Identity

We are given that \(\sin(\theta) = \frac{3}{5}\) and that \(\theta\) lies in the interval \(0 < \theta < \frac{\pi}{2}\). Our goal is to calculate the value of \(\sin(2\theta)\). To achieve this, we will use the double-angle identity for sine: \[\sin(2\theta) = 2 \sin(\theta) \cos(\theta).\] Since we already know the value of \(\sin(\theta)\), we only need to determine the value of \(\cos(\theta)\).

  1. Calculating the Value of Cosine

To find \(\cos(\theta)\), we use the Pythagorean identity: \[\sin^2(\theta) + \cos^2(\theta) = 1.\] Substituting the given value \(\sin(\theta) = \frac{3}{5}\), we have \[\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1.\] This simplifies to \[\frac{9}{25} + \cos^2(\theta) = 1.\] Solving for \(\cos^2(\theta)\): \[\cos^2(\theta) = 1 - \frac{9}{25} = \frac{25-9}{25} = \frac{16}{25}.\] Since \(0 < \theta < \frac{\pi}{2}\), the cosine function is positive in this first quadrant. Therefore, we take the positive square root: \[\cos(\theta) = \sqrt{\frac{16}{25}} = \frac{4}{5}.\]

  1. Applying the Double-Angle Identity and Conclusion

Now that we have both \(\sin(\theta) = \frac{3}{5}\) and \(\cos(\theta) = \frac{4}{5}\), we can substitute these values into the double-angle identity: \[\sin(2\theta) = 2 \sin(\theta) \cos(\theta).\] Substituting the values: \[\sin(2\theta) = 2 \times \left(\frac{3}{5}\right) \times \left(\frac{4}{5}\right).\] Performing the multiplication: \[\sin(2\theta) = 2 \times \frac{12}{25} = \frac{24}{25}.\] Thus, the exact value of \(\sin(2\theta)\) is \(\frac{24}{25}\).

Examples

The double-angle identity is a fundamental concept in trigonometry, essential for simplifying complex expressions and solving real-world problems. For example, in physics, when calculating the horizontal range of a projectile fired at an angle \(\theta\), the formula involves \(\sin(2\theta)\). Knowing the trigonometric ratios for a single angle allows us to precisely determine motion characteristics without needing the double angle explicitly measured. This technique is vital in fields like engineering, architecture, and computer graphics where trigonometric relationships govern spatial geometry and wave patterns.

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