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Chapter 4: Problem 81

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\cos \frac{23 \pi}{4}$$

Short Answer

Expert verified
The exact value of \(\cos(23\pi/4)\) is \(-\sqrt{2}/2\).

Step by step solution

01

Express the Angle in Terms of Integer Multiples of \(2\pi\)

First, divide the numerator of the fraction \(23\) by \(4\) in order to express \(23\pi/4\) as the sum of an integer multiple of \(2\pi\) and an angle between \(0\) and \(2\pi\). The quotient is \(5\), and the remainder is \(3\), so \(23\pi/4\) can be expressed as \(5 \times 2\pi + 3\pi/4\).
02

Determine the Reference Angle

Then, recognize that the cosine function has period \(2\pi\), so \(\cos(5 \times 2\pi + 3\pi/4) = \cos(3\pi/4)\). The angle \(\pi/4\) is the reference angle for \(\pi/4\), \(\pi - \pi/4 = 3\pi/4\), \( \pi + \pi/4 = 5\pi/4\), and \(2\pi - \pi/4 = 7\pi/4\), so the reference angle for \(3\pi/4\) is \(\pi/4\).
03

Find the Cosine of the Reference Angle

Once the reference angle \( \pi/4\) is found, find the cosine of this reference angle. From the definition of cosine, \(\cos(\pi/4) = \sqrt{2}/2\). However, for an angle in the second quadrant like \(3\pi/4\), cosine is negative.

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