1. **State the problem:** We need to find the exact value of the function $$f(x) = \sin(x) + 3 \tan(x)$$ at $$x = \frac{2\pi}{3}$$. 2. **Recall the definitions and values:** - $$\sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$$. - $$\tan\left(\frac{2\pi}{3}\right) = \tan\left(\pi - \frac{\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}$$. 3. **Substitute these values into the function:** $$f\left(\frac{2\pi}{3}\right) = \sin\left(\frac{2\pi}{3}\right) + 3 \tan\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} + 3(-\sqrt{3})$$ 4. **Simplify the expression:** $$\frac{\sqrt{3}}{2} - 3\sqrt{3} = \frac{\sqrt{3}}{2} - \frac{6\sqrt{3}}{2} = -\frac{5\sqrt{3}}{2}$$ 5. **Final answer:** $$f\left(\frac{2\pi}{3}\right) = -\frac{5\sqrt{3}}{2}$$ This is the exact value of the function at the given point.