/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 66 Find an exact expression for \(\... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 66

Find an exact expression for \(\cos \frac{\pi}{16}\).

Short Answer

Expert verified
The exact expression for \(\cos{\frac{\pi}{16}}\) is: \[\cos{\frac{\pi}{16}} = \sqrt{\frac{1 + \sqrt{\frac{1 + \sqrt{\frac{1}{2}}}{2}}}{2}}\]

Step by step solution

01

Start with \(\cos{\frac{\pi}{2}}\)

Given that \(\cos{\frac{\pi}{16}}\) is the value we want to find, we'll start with \(\cos{\frac{\pi}{2}} = 0\), which is an angle value that we know.
02

Apply the half-angle identity

To find \(\cos{\frac{\pi}{4}}\), use the half-angle identity with \(\cos{\frac{\pi}{2}} = 0\): \[\cos{\frac{\pi}{4}} = \pm\sqrt{\frac{1 + \cos{\frac{\pi}{2}}}{2}}\] Since the cosine is positive in the first quadrant, we use the positive square root: \[\cos{\frac{\pi}{4}} = \sqrt{\frac{1 + 0}{2}} = \sqrt{\frac{1}{2}}\]
03

Apply the half-angle identity one more time

Now, use the half-angle identity again to find \(\cos{\frac{\pi}{8}}\): \[\cos{\frac{\pi}{8}} = \pm\sqrt{\frac{1 + \cos{\frac{\pi}{4}}}{2}} = \sqrt{\frac{1 + \sqrt{\frac{1}{2}}}{2}}\]
04

Apply the half-angle identity one last time

Finally, use the half-angle identity to find \(\cos{\frac{\pi}{16}}\): \[\cos{\frac{\pi}{16}} = \pm\sqrt{\frac{1 + \cos{\frac{\pi}{8}}}{2}}\] Again, since \(\frac{\pi}{16}\) is in the first quadrant, we use the positive square root: \[\cos{\frac{\pi}{16}} = \sqrt{\frac{1 + \sqrt{\frac{1 + \sqrt{\frac{1}{2}}}{2}}}{2}}\]

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Most popular questions from this chapter

Show that $$\sin ^{2}(2 \theta)=4\left(\sin ^{2} \theta-\sin ^{4} \theta\right)$$ for all \(\theta\).

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