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Chapter 5: Problem 2

For \(\theta=5\) radians, evaluate each of the following: (a) \(\cos ^{2} \theta\) (b) \(\cos \left(\theta^{2}\right)\)

Short Answer

Expert verified
(a) \(\cos^2\theta \approx 0.91932859\) (b) \(\cos(\theta^2) \approx 0.99120281\)

Step by step solution

01

Find the cosine of \(\theta\)

First, we need to find the cosine of \(\theta\) which is equal to \(\cos(5)\). You can use a calculator or any tool to find the cosine of 5 radians. The value of \(\cos(5)\) is approximately -0.95892427.
02

Calculate \(\cos^2\theta\)

Now that we have the cosine of \(\theta\), we can calculate \(\cos^2\theta\) as follows: \[\cos^2\theta = \left(\cos\theta\right)^2\] Plug in the value we found for \(\cos\theta\): \[\cos^2\theta = (-0.95892427)^2\] Now, evaluate the square: \[\cos^2\theta \approx 0.91932859\]
03

Calculate \(\cos(\theta^2)\)

Next, we need to calculate \(\cos(\theta^2)\). First, let's find the value of \(\theta^2\): \[\theta^2 = (5)^2 = 25\] Now, we will find the cosine of \(25\) radians, which is \(\cos(25)\): \[\cos(\theta^2) = \cos(25)\] Using a calculator or any tool, we find that \(\cos(25)\) is approximately 0.99120281. So the final answers are: (a) \(\cos^2\theta \approx 0.91932859\) (b) \(\cos(\theta^2) \approx 0.99120281\)

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