Problem 68 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\cos \frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \(\cos \frac{3 \pi}{4}\) is \(-\frac{\sqrt{2}}{2}\)

Step 1: Map the given angle on the unit circle

Firstly, recognize that \(\frac{3 \pi}{4}\) is in the second quadrant on the unit circle since it is larger than \(\frac{\pi}{2}\) but less than \(\pi\). In the second quadrant, cosine is negative.

Step 2: Determine the reference angle

The reference angle is the acute angle the terminal side of \(\frac{3 \pi}{4}\) makes with the x-axis. It is calculated as: Ref angle = \(\pi - \frac{3 \pi}{4} = \frac{\pi}{4}\)

Step 3: Compute the cosine of the reference angle

The reference angle \(\frac{\pi}{4}\) is well known and it's exact values can be computed without a calculator. \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)

Step 4: Determine cosine of the original angle

Since we’re in the second quadrant, and cosine is negative in the second quadrant, the cosine of \(\frac{3 \pi}{4}\) would be the negative of the cosine of \(\frac{\pi}{4}\). Thus, \(\cos \frac{3 \pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}\)