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Chapter 4: Problem 92

In Exercises \(87-92\), find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$\sin \frac{3 \pi}{2} \tan \left(-\frac{8 \pi}{3}\right)+\cos \left(-\frac{5 \pi}{6}\right)$$

Short Answer

Expert verified
-\sqrt{3}/2

Step by step solution

01

Evaluate Individual Expressions

The first step will be to break the expression into separate parts and find the values from the unit circle. Note that the angles provided are in multiples of \(\pi\), signifying that we're working in radians.
02

Evaluate \(\sin \frac{3 \pi}{2}\)

The sine of \(\frac{3 \pi}{2}\) is -1. This corner of the unit circle lies on the negative y axis, and its y-coordinate is -1.
03

Evaluate \(\tan \left(-\frac{8 \pi}{3}\right)\)

The tangent function has a period of \(\pi\), which means \( \tan \left(-\frac{8 \pi}{3}\right) = \tan \left(-\frac{8 \pi}{3} + \frac{8 \pi}{3}\right) = \tan \left(0\right) = 0\). Here we're using the periodic property of the tangent function.
04

Evaluate \(\cos \left(-\frac{5 \pi}{6}\right)\)

The cosine function is even, which means \(\cos \left(-\theta\right) = \cos \left(\theta\right)\). Thus, \(\cos \left(-\frac{5 \pi}{6}\right) = \cos \left(\frac{5 \pi}{6}\right)\), which is equal to -\(\sqrt{3}/2\). This is obtained from standard values of cosine on the unit circle.
05

Put Everything Together

Now substitute the values obtained in the given expression: \(\sin \frac{3 \pi}{2} \tan \left(-\frac{8 \pi}{3}\right)+\cos \left(-\frac{5 \pi}{6}\right) = (-1 * 0) + (-\sqrt{3}/2) = -\sqrt{3}/2\)

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