1. The problem is to find the value of $\sin(3\theta)$ without using the triple angle formula. 2. We start with the angle addition formula for sine: $$\sin(a+b) = \sin a \cos b + \cos a \sin b$$ 3. Express $\sin(3\theta)$ as $\sin(2\theta + \theta)$. 4. Apply the angle addition formula: $$\sin(3\theta) = \sin(2\theta) \cos(\theta) + \cos(2\theta) \sin(\theta)$$ 5. Use the double angle formulas: $$\sin(2\theta) = 2 \sin \theta \cos \theta$$ and $$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$$ 6. Substitute these into the expression: $$\sin(3\theta) = (2 \sin \theta \cos \theta) \cos \theta + (\cos^2 \theta - \sin^2 \theta) \sin \theta$$ 7. Simplify: $$\sin(3\theta) = 2 \sin \theta \cos^2 \theta + \sin \theta \cos^2 \theta - \sin^3 \theta$$ 8. Combine like terms: $$\sin(3\theta) = 3 \sin \theta \cos^2 \theta - \sin^3 \theta$$ 9. Use the Pythagorean identity $\cos^2 \theta = 1 - \sin^2 \theta$ to write: $$\sin(3\theta) = 3 \sin \theta (1 - \sin^2 \theta) - \sin^3 \theta$$ 10. Expand: $$\sin(3\theta) = 3 \sin \theta - 3 \sin^3 \theta - \sin^3 \theta = 3 \sin \theta - 4 \sin^3 \theta$$ Final answer: $$\sin(3\theta) = 3 \sin \theta - 4 \sin^3 \theta$$