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Chapter 4: Problem 87

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

Short Answer

Expert verified
Hence the exact value of the given expression is \(-\frac{\sqrt{3}}{2} + \frac{1}{2}\)

Step by step solution

01

Recognize the Form

The given problem seems to be in the form of a product-to-sum formula, which is \(\sin (A) \cos (B) - \cos (A) \sin (B)\), that simplifies to \(\sin (A - B)\). Here, \(A = \frac{\pi}{3}\) and \(B = \pi \& \frac{3\pi}{2}\) respectively.
02

Use the Trigonometric Identity

Substituting the values into the identity, we get \(\sin (\frac{\pi}{3} - \pi) - \sin (\frac{\pi}{3} - \frac{3\pi}{2})\). This simplifies to \(\sin (-\frac{2\pi}{3}) - \sin (-\frac{5\pi}{6})\).
03

Find the sine values

On the unit circle, the sine value at \(-\frac{2\pi}{3}\) and \(-\frac{5\pi}{6}\) are \(-\frac{\sqrt{3}}{2}\) and \(-\frac{1}{2}\) respectively.
04

Result

After substituting these trigonometric values the expression becomes \(-\frac{\sqrt{3}}{2} - \left(-\frac{1}{2}\right)\) which simplifies to \(-\frac{\sqrt{3}}{2} + \frac{1}{2}\).

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