/*! 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 6 The next two exercises emphasize... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 6

The next two exercises emphasize that \(\cos \frac{\theta}{2}\) does not equal \(\frac{\cos \theta}{2}\). For \(\theta=-80^{\circ},\) evaluate each of the following: (a) \(\cos \frac{\theta}{2}\) (b) \(\frac{\cos \theta}{2}\)

Short Answer

Expert verified
(a) \(\cos \frac{\theta}{2} = 0.76604444\) (b) \(\frac{\cos \theta}{2} = 0.49240388\) The results show that \(\cos \frac{\theta}{2}\) is not equal to \(\frac{\cos \theta}{2}\) for the given angle \(\theta=-80^{\circ}\).

Step by step solution

01

Find the value of cos(-80掳)

First, let's find the value of \(\cos (-80^{\circ})\): Using a calculator or a trigonometric table, we find that \[\cos (-80^{\circ}) = 0.98480775\]
02

Evaluate \(\cos \frac{\theta}{2}\) for \(\theta=-80^{\circ}\)

Now that we have the value for \(\cos (-80^{\circ})\), we can evaluate the expression \(\cos \frac{\theta}{2}\): \[\cos \frac{\theta}{2} = \cos \frac{-80^{\circ}}{2} = \cos (-40^{\circ})\] Using a calculator or a trigonometric table, we find that \[\cos (-40^{\circ}) = 0.76604444\] So, \[\cos \frac{\theta}{2} = 0.76604444\]
03

Evaluate \(\frac{\cos \theta}{2}\) for \(\theta=-80^{\circ}\)

Now let's evaluate the expression \(\frac{\cos \theta}{2}\): Using the value for \(\cos (-80^{\circ})\) we found earlier, we have \[\frac{\cos \theta}{2} = \frac{\cos (-80^{\circ})}{2} = \frac{0.98480775}{2} = 0.49240388\] So, \[\frac{\cos \theta}{2} = 0.49240388\]
04

Compare the results (a) and (b)

Finally, let's compare both results: (a) \(\cos \frac{\theta}{2} = 0.76604444\) (b) \(\frac{\cos \theta}{2} = 0.49240388\) As we can see, the value of \(\cos \frac{\theta}{2}\) is not equal to \(\frac{\cos \theta}{2}\), thus emphasizing that the statement \(\cos \frac{\theta}{2} \neq \frac{\cos \theta}{2}\) is true for the given angle \(\theta=-80^{\circ}\).

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