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Chapter 4: Problem 52

Use the properties of inverse trigonometric functions to evaluate the expression. $$\arccos \left(\cos \frac{7 \pi}{2}\right)$$

Short Answer

Expert verified
The value of the expression \( \arccos \left( \cos \frac{7 \pi}{2} \right) \) is \( \pi \).

Step by step solution

01

Understanding Cosine Function

The cosine function, denoted as \( \cos \), is a periodic function with a period of \(2 \pi \). This means if you add or subtract any multiple of \(2 \pi \) to the argument of a cosine function, the function's value remains unchanged. \( \cos (\theta + 2 \pi k) = \cos \theta \) where \( k \) is an integer.
02

Simplifying the Cosine Argument

Using the property established in Step 1, the argument of \(\cos\), \(\frac{7 \pi}{2}\), can be reduced to an equivalent value within the range \([0, 2 \pi ]\). \(\frac{7 \pi}{2}\) = \(\frac{6 \pi}{2}\) + \(\frac{1\pi}{2}\) = \(3\pi + \frac{\pi}{2}\). As the cosine function repeats every \(2 \pi\), \(3 \pi\) is equivalent to \(\pi\), and \(\frac{\pi}{2}\) remains as it is. So \(\frac{7 \pi}{2}\)'s cosine-equivalent in the range \([0, 2 \pi ]\) is \(\pi + \frac{\pi}{2}\) = \(\frac{3 \pi}{2}\).
03

Evaluating Cosine and Arccosine

Substituting \( \frac{3 \pi}{2} \) back into the original expression gives: \( \arccos \left( \cos \left( \frac{3 \pi}{2} \right) \right) \). The value of \( \cos \frac{3 \pi}{2} \) is -1. So, the expression simplifies to \( \arccos (-1) \). By definition, the arccosine function (i.e., the inverse cosine function) tells you what angle has a given cosine value. The angle which has -1 as its cosine value is \( \pi \).

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Most popular questions from this chapter

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{2} \cos 20 \pi t$$

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