Find the exact value of the following expression without using a calculator.Check your answer with a calculator. (sin(frac (5pi )/(6)))(cos((5pi )/(6))) (sin(frac (5pi )/(6)))(cos((5pi )/(6)))=square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression )

To find the exact value of the expression \(\frac{\sin\left(\frac{5\pi}{6}\right)}{\cos\left(\frac{5\pi}{6}\right)}\), we can use the trigonometric identity for the tangent function:<br /><br />\[<br />\tan(x) = \frac{\sin(x)}{\cos(x)}<br />\]<br /><br />So, we have:<br /><br />\[<br />\frac{\sin\left(\frac{5\pi}{6}\right)}{\cos\left(\frac{5\pi}{6}\right)} = \tan\left(\frac{5\pi}{6}\right)<br />\]<br /><br />Now, let's find the value of \(\tan\left(\frac{5\pi}{6}\right)\).<br /><br />The angle \(\frac{5\pi}{6}\) is in the second quadrant, where the tangent function is negative. We can use the reference angle \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\) to find the value of \(\tan\left(\frac{5\pi}{6}\right)\).<br /><br />We know that:<br /><br />\[<br />\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}<br />\]<br /><br />Since \(\frac{5\pi}{6}\) is in the second quadrant, where the tangent function is negative, we have:<br /><br />\[<br />\tan\left(\frac{5\pi}{6}\right) = -\frac{1}{\sqrt{3}}<br />To rational, we multiply the numerator and denominator by \(\sqrt{3}\):<br /><br />\[<br />\tan\left(\frac{5\pi}{6}\right) = -\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}<br />\]<br /><br />Therefore, the exact value of the expression \(\frac{\sin\left(\frac{5\pi}{6}\right)}{\cos\left(\frac{5\pi}{6}\right)}\) is:<br /><br />\[<br />\boxed{-\frac{\sqrt{3}}{3}}<br />\]