91Ó°ÊÓ

Firstly, recognize the angles in the problem and calculate their exact values. We have three trigonometric functions to work out: \n\n1. \(\sin \frac{3 \pi}{2}\)\n2. \(\tan \left(-\frac{15 \pi}{4}\right)\)\n3. \(\cos \left(-\frac{5 \pi}{3}\right)\)\n\nFrom the unit circle, we know that \(\sin \frac{3 \pi}{2} = -1\), because at \(\frac{3 \pi}{2}\) radians, the y-coordinate is -1. \n\nThe function \(\tan x\) is periodic with period \(\pi\), therefore, \(\tan \left(-\frac{15 \pi}{4}\right) = \tan \left(-\frac{15 \pi}{4} + 4\pi\right) = \tan \frac{\pi}{4} = 1\), because at \(\frac{\pi}{4}\) radians, both the x and y coordinates are \(\frac{1}{\sqrt{2}}\), and hence their ratio is 1.\n\nAgain, from the unit circle, we know that \(\cos \left(-\frac{5 \pi}{3}\right) = \cos \left(-\frac{5 \pi}{3} + 2\pi\right) = \cos \frac{\pi}{3} = \frac{1}{2}\), because at \(\frac{\pi}{3}\) radians, the x-coordinate is \(\frac{1}{2}\).

Now that we have found the exact values of each trigonometric function, they can be substituted back into the original expression. This expression becomes: \n\n-\(1*1 - \frac{1}{2}\) = -1-\frac{1}{2}\)

The last step is to simplify the negative mixed number -1-\(\frac{1}{2}\) to a single fraction. We can do this by multiplying the whole part of the mixed number by the denominator of the fraction and then adding the numerator. After lying out the fraction, the resulting fraction is -\(\frac{3}{2}\).