Chapter 10: Q .49. (page 812)

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

Short Answer

Expert verified

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

Step by step solution

Step 1:Given information

$Consider the direction cosines \cos α = \frac{1}{2}, \cos β = \frac{1}{2} .$

Step 2:Simplification

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

$\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$Also, \cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$Now, since \cos α = \frac{1}{2}, \cos β = \frac{1}{2} .$

$Substitute the values in (1)$

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

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

$\frac{2}{4} + \cos^{2} γ = 1$

$\frac{1}{2} + \cos^{2} γ = 1$

$\cos^{2} γ = 1 - \frac{1}{2}$

$\cos γ = \pm \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$

$= \pm \frac{\sqrt{2}}{2}$

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

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

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

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In Exercises 49-51, two direction cosines are given. Use Exercise 48 to find the third direction cosine.cosα=12,cosβ=12.

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Calculus

Theoretical and Mathematical Physics

Statistics

Geometry

Study anywhere. Anytime. Across all devices.

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