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Chapter 10: Q .51. (page 812)

In Exercises 49-51, two direction cosines are given. Use Exercise 48 to find the third direction cosine.
$51. \cos 伪 = \frac{1}{2}, \cos 纬 = \frac{\sqrt{3}}{4} .$

Short Answer

Expert verified

$the value of the third directicn cosine is 卤 \frac{3}{4}$

Step by step solution

01

Step 1:Given information

$\cos 伪 = \frac{1}{2}, \cos 纬 = \frac{\sqrt{3}}{4} .$

02

Step 2:Simplification

$Consider the drection cosines \cos 伪 = \frac{1}{2}, \cos 纬 = \frac{\sqrt{3}}{4} .$

$The objective is to fnd the third direction cosine using the result Consider the direction cosinas$

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

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

width="234" height="45" role="math">Nos,sincecos伪=12,cos纬=34

$Subetituse the values in (1)$

${(\frac{1}{2})}^{2} + \cos^{2} 尾 + {(\frac{\sqrt{3}}{4})}^{2} = 1$

$\frac{1}{4} + \cos^{2} 尾 + \frac{3}{16} = 1$

$\frac{4 + 3}{16} + \cos^{2} 尾 = 1$

$\frac{7}{16} + \cos^{2} 尾 = 1$

$\cos^{2} 尾 = 1 - \frac{7}{16}$

$= \frac{16 - 7}{16}$ $= 卤 \frac{9}{16}$

$\cos 纬 = 卤 \sqrt{\frac{9}{16}}$

$= 卤 \frac{3}{4}$

$the value of the third directicn cosine is 卤 \frac{3}{4}]$

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