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# Solution for If a Line Makes Angles α, β and γ with the Coordinate Axes, Find the Value of Cos 2α + Cos 2β + Cos 2γ. - CBSE (Science) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

#### Question

If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.

#### Solution

$\text{ It is given that the the line makes angles } \alpha, \beta, \gamma \text{ with the coordinate axis }.$

$\therefore l = \cos \alpha, m = \cos \beta \text{ and } n = \cos \gamma$

$\Rightarrow l^2 + m^2 + n^2 = 1$

$\Rightarrow \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 . . . \left( 1 \right)$

$\text{ Now} ,$

$\cos 2\alpha + \cos 2\beta + \cos 2\gamma = \left( 2 \cos^2 \alpha - 1 \right) + \left( 2 \cos^2 \beta - 1 \right) + \left( 2 \cos^2 \gamma - 1 \right)$

$= 2\left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right) - 3$

$= 2\left( 1 \right) - 3 \left [ \text{ From }\left( 1\right) \right]$

$= - 1$

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#### Video TutorialsVIEW ALL [3]

Solution If a Line Makes Angles α, β and γ with the Coordinate Axes, Find the Value of Cos 2α + Cos 2β + Cos 2γ. Concept: Direction Cosines and Direction Ratios of a Line.
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