Find the exact values of the six trigonometric functions of the given angle. Do not use a calculator.
\[
\frac{5 \pi}{3}
\]

Step 1 ：The angle \(\frac{5 \pi}{3}\) is in standard position and it is a positive angle measured counterclockwise from the positive x-axis. The angle \(\frac{5 \pi}{3}\) is equivalent to \(\frac{\pi}{3}\) because \(5 \pi/3 = 2 \pi + \pi/3\), and \(2 \pi\) is a full rotation which brings us back to the same position.

Step 2 ：The six trigonometric functions of \(\frac{\pi}{3}\) are well known. They are: \[\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\], \[\cos(\frac{\pi}{3}) = \frac{1}{2}\], \[\tan(\frac{\pi}{3}) = \sqrt{3}\], \[\csc(\frac{\pi}{3}) = \frac{2}{\sqrt{3}}\], \[\sec(\frac{\pi}{3}) = 2\], \[\cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}}\]

Step 3 ：So, the six trigonometric functions of \(\frac{5 \pi}{3}\) are the same as those of \(\frac{\pi}{3}\).

Step 4 ：Final Answer: \[\sin\left(\frac{5 \pi}{3}\right) = \boxed{\frac{\sqrt{3}}{2}}, \quad \cos\left(\frac{5 \pi}{3}\right) = \boxed{\frac{1}{2}}, \quad \tan\left(\frac{5 \pi}{3}\right) = \boxed{\sqrt{3}}\], \[\csc\left(\frac{5 \pi}{3}\right) = \boxed{\frac{2}{\sqrt{3}}}, \quad \sec\left(\frac{5 \pi}{3}\right) = \boxed{2}, \quad \cot\left(\frac{5 \pi}{3}\right) = \boxed{\frac{1}{\sqrt{3}}}\]