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Chapter 10: Q .49. (page 812)

In Exercises 49-51, two direction cosines are given. Use Exercise 48 to find the third direction cosine.
$\cos 伪 = \frac{1}{2}, \cos 尾 = \frac{1}{2} .$

Short Answer

Expert verified

$The value of the third direction cosine is 卤 \frac{\sqrt{2}}{2} .$

Step by step solution

01

Step 1:Given information

$Consider the direction cosines \cos 伪 = \frac{1}{2}, \cos 尾 = \frac{1}{2} .$

02

Step 2:Simplification

$The objective is to find the third direction cosine using the result Consider the direction cosines$

$\cos^{2} 伪 + \cos^{2} 尾 + \cos^{2} 纬 = 1$

$Also, \cos^{2} 伪 + \cos^{2} 尾 + \cos^{2} 纬 = 1$

$Now, since \cos 伪 = \frac{1}{2}, \cos 尾 = \frac{1}{2} .$

$Substitute the values in (1)$

${(\frac{1}{2})}^{2} + {(\frac{1}{2})}^{2} + \cos^{2} 纬 = 1$

$\frac{1}{4} + \frac{1}{4} + \cos^{2} 纬 = 1$

$\frac{2}{4} + \cos^{2} 纬 = 1$

$\frac{1}{2} + \cos^{2} 纬 = 1$

$\cos^{2} 纬 = 1 - \frac{1}{2}$

$\cos 纬 = 卤 \frac{1}{\sqrt{2}} 脳 \frac{\sqrt{2}}{\sqrt{2}}$

$= 卤 \frac{\sqrt{2}}{2}$

$Hence, the value of the third direction cosine is 卤 \frac{\sqrt{2}}{2} .$

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