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Chapter 4: Problem 90

In Exercises \(87-92\), find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$\sin \frac{17 \pi}{3} \cos \frac{5 \pi}{4}+\cos \frac{17 \pi}{3} \sin \frac{5 \pi}{4}$$

Short Answer

Expert verified
So, the exact value of the given expression is \(\frac{\sqrt{2}}{2}\).

Step by step solution

01

Simplify angles

Because \(\sin\) and \(\cos\) are periodic with a period of \(2\pi\), we can simplify the given angles. For instance, \(\sin \frac{17\pi}{3}\) is equivalent to \(\sin \pi\), because \(\frac{17\pi}{3}\) is equivalent to \(7\pi + \pi\). Similarly, \(\cos \frac{17\pi}{3}\) is equivalent to \(\cos \pi\). Now the expression becomes \(\sin \pi \cos \frac{5\pi}{4} + \cos \pi \sin \frac{5\pi}{4}\).
02

Compute the sine and cosine values

Recall that \(\sin \pi = 0\) and \(\cos \pi = -1\). Further, \(\cos \frac{5\pi}{4}\) and \(\sin \frac{5\pi}{4}\) are both \(-\frac{1}{\sqrt{2}}\). Substituting these values in, we get \(0 \times -\frac{1}{\sqrt{2}} - 1 \times -\frac{1}{\sqrt{2}}\).
03

Simplify the expression

The expression simplifies to \(\frac{1}{\sqrt{2}}\), which can also be written as \(\frac{\sqrt{2}}{2}\) by rationalizing the denominator.

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Most popular questions from this chapter

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-4 \cos \left(2 x-\frac{\pi}{2}\right)$$

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