/*! 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 5 The next two exercises emphasize... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 5

The next two exercises emphasize that \(\cos \frac{\theta}{2}\) does not equal \(\frac{\cos \theta}{2}\). For \(\theta=6\) radians, evaluate each of the following: (a) \(\cos \frac{\theta}{2}\) (b) \(\frac{\cos \theta}{2}\)

Short Answer

Expert verified
Using a calculator, we find: (a) \(\cos \left(\frac{6}{2}\right) = \cos(3) \approx -0.9900\) (b) \(\frac{\cos(6)}{2} \approx \frac{0.9602}{2} = 0.4801\) The values are clearly not equal, demonstrating that \(\cos \left(\frac{\theta}{2}\right) \neq \frac{\cos \theta}{2}\).

Step by step solution

01

Substitute the given angle 胃 into the expressions

We have 胃 = 6 radians. Now, we will substitute the angle into both expressions: (a) \(\cos \left(\frac{\theta}{2}\right)\) (b) \(\frac{\cos \theta}{2}\)
02

Evaluate expression (a)

In order to evaluate (a) \(\cos \left(\frac{\theta}{2}\right)\), we need to find the cosine of half of the angle 胃. \[\cos \left(\frac{6}{2}\right) = \cos (3)\] Now, we can use a calculator to obtain the numeric value of \(\cos(3)\) radians.
03

Evaluate expression (b)

In order to evaluate (b) \(\frac{\cos \theta}{2}\), we need to find the cosine of the angle 胃 and then divide it by 2. \[\frac{\cos(6)}{2}\] Now use a calculator to obtain the numeric value of \(\cos(6)\) radians and then divide it by 2.
04

Compare the results of expressions (a) and (b)

After calculating the numeric values for both (a) and (b) expressions, we should notice that they are not equal. This demonstrates that \(\cos \left(\frac{\theta}{2}\right)\) does not equal \(\frac{\cos \theta}{2}\) as we were tasked to emphasizing.

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