1. **State the problem:** We need to determine the exact values of $\sin\left(\frac{2}{3}\pi\right)$ and $\cos\left(\frac{2}{3}\pi\right)$ and check if the given decimal approximations $0.0329$ and $0.999$ are correct. 2. **Recall the formulas and important rules:** - The sine and cosine of an angle $\theta$ in radians are defined on the unit circle. - $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$. - $\pi$ radians equals 180 degrees. 3. **Evaluate $\sin\left(\frac{2}{3}\pi\right)$:** - Note that $\frac{2}{3}\pi = 120^\circ$. - Using the identity $\sin\left(\pi - x\right) = \sin x$, we get: $$\sin\left(\frac{2}{3}\pi\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right)$$ - We know $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$. 4. **Evaluate $\cos\left(\frac{2}{3}\pi\right)$:** - Using the identity $\cos\left(\pi - x\right) = -\cos x$, we get: $$\cos\left(\frac{2}{3}\pi\right) = \cos\left(\pi - \frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right)$$ - We know $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$, so: $$\cos\left(\frac{2}{3}\pi\right) = -\frac{1}{2}$$ 5. **Compare with given decimal values:** - $\sin\left(\frac{2}{3}\pi\right) = \frac{\sqrt{3}}{2} \approx 0.866$, which is not $0.0329$. - $\cos\left(\frac{2}{3}\pi\right) = -\frac{1}{2} = -0.5$, which is not $0.999$. **Final answers:** $$\sin\left(\frac{2}{3}\pi\right) = \frac{\sqrt{3}}{2}$$ $$\cos\left(\frac{2}{3}\pi\right) = -\frac{1}{2}$$