Problem 39 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\cos \frac{9 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \(\cos(\frac{9 \pi}{4})\) is \(\frac{\sqrt{2}}{2}\).

Step 1: Simplify the Fraction

First, simplify the given fraction. As \(\frac{9 \pi}{4} = 2 \pi + \frac{1 \pi}{4}\) (which is more than a full rotation of \(2\pi\)), subtract full rotations from \(\frac{9 \pi}{4}\) until you get an angle that is between 0 and \(2\pi\). So, \(\frac{9 \pi}{4} - 2 \pi = \frac{9 \pi}{4} - \frac{8 \pi}{4} = \frac{\pi}{4}\).

Step 2: Find the Cosine Value

Now, find the cosine value of \(\frac{\pi}{4}\). The unit circle tells us that \(\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\).

Step 3: Conclusion

So the exact value of \(\cos(\frac{9 \pi}{4})\) is \(\frac{\sqrt{2}}{2}\).