Let $f(x)=\sin x$. Find the exact value of the following expression. Do not use a calculator. \[ f\left(\frac{2 \pi}{3}-\frac{\pi}{6}\right)+f\left(\frac{2 \pi}{3}\right)-f\left(\frac{\pi}{6}\right) \]

Step 1 ：Given the function \(f(x)=\sin x\), we are asked to find the exact value of the expression \(f\left(\frac{2 \pi}{3}-\frac{\pi}{6}\right)+f\left(\frac{2 \pi}{3}\right)-f\left(\frac{\pi}{6}\right)\).

Step 2 ：First, we simplify the arguments of the function. \(\frac{2 \pi}{3}-\frac{\pi}{6}\) simplifies to \(\frac{\pi}{2}\).

Step 3 ：So, the expression becomes \(f\left(\frac{\pi}{2}\right)+f\left(\frac{2 \pi}{3}\right)-f\left(\frac{\pi}{6}\right)\).

Step 4 ：Substituting these values into the function \(f(x)=\sin x\), we get \(\sin\left(\frac{\pi}{2}\right)+\sin\left(\frac{2 \pi}{3}\right)-\sin\left(\frac{\pi}{6}\right)\).

Step 5 ：Using the unit circle, we know that \(\sin\left(\frac{\pi}{2}\right)=1\), \(\sin\left(\frac{2 \pi}{3}\right)=\frac{\sqrt{3}}{2}\), and \(\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}\).

Step 6 ：Substituting these values in, we get \(1+\frac{\sqrt{3}}{2}-\frac{1}{2}\).

Step 7 ：Simplifying this expression, we get \(\frac{1}{2} + \frac{\sqrt{3}}{2}\).

Step 8 ：Final Answer: The exact value of the expression \(f\left(\frac{2 \pi}{3}-\frac{\pi}{6}\right)+f\left(\frac{2 \pi}{3}\right)-f\left(\frac{\pi}{6}\right)\) is \(\boxed{\frac{1}{2} + \frac{\sqrt{3}}{2}}\).