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Chapter 5: Problem 68

Find a formula for \(\cos (4 \theta)\) in terms of \(\cos \theta\).

Short Answer

Expert verified
The formula for \(\cos(4 \theta)\) in terms of \(\cos \theta\) is: \[\cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 1\]

Step by step solution

01

Use the double-angle identity for \(\cos(2\theta)\)

First, we need to find the expression for \(\cos(2\theta)\) in terms of \(\cos(\theta)\). Using the double-angle identity, we get: \[\cos(2\theta) = 2\cos^2(\theta) - 1\]
02

Use the double-angle identity for \(\cos(4\theta)\)

Now, we want to find \(\cos(4\theta)\). We can notice that \(4\theta = 2*(2\theta)\), so we can use the double-angle identity again with \(\alpha = 2\theta\): \[\cos(4\theta) = \cos(2(2\theta)) = 2\cos^2(2\theta) - 1\]
03

Replace \(\cos(2\theta)\) with the expression in terms of \(\cos\theta\)

We already found \(\cos(2\theta) = 2\cos^2(\theta) - 1\), so now we can substitute this expression into our formula for \(\cos(4\theta)\): \[\cos(4\theta) = 2\left(2\cos^2(\theta) - 1\right)^2 - 1\]
04

Simplify the formula

Now, we need to simplify the formula for \(\cos(4\theta)\). Start by expanding the square and then multiply and combine the terms: \[\cos(4\theta) = 2(4\cos^4(\theta) - 4\cos^2(\theta) + 1) - 1\] \[\cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 2 - 1\] \[\cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 1\] Now, we have found a formula for \(\cos(4 \theta)\) in terms of \(\cos \theta\): \[\cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 1\]

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Most popular questions from this chapter

Find an exact expression for \(\sin 15^{\circ}\).

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