A line makes angles α , β , γ with the coordinate axes. If α + β = π 2 , then cos α + cos β + cos γ 2 is equal to

We have, cos 2 α + cos 2 β + cos 2 γ = 1 ⇒ cos 2 α + cos 2 π 2 - α + cos 2 γ = 1 [given α + β = π 2 ] ⇒ cos 2 α + sin 2 α + cos 2 γ = 1 ⇒ 1 + cos 2 γ = 1 ⇒ cos 2 γ = 0 ⇒ cos γ = 0 ∴ cos α + cos β + cos γ 2 = cos α + sin α 2 = 1 + 2 sin α · cos α = 1 + sin 2 α

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Mathematics3D-Coordinate GeometryCoordinates of a point in space, Direction cosines,Easy2 minPYQ_2018

MathematicsEasysingle choice

A line makes angles $α, β, γ$ with the coordinate axes. If $α + β = \frac{π}{2}$ , then ${(\cos α + \cos β + \cos γ)}^{2}$ is equal to

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Answer:

Solution:

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

$\Rightarrow \cos^{2} α + \cos^{2} (\frac{π}{2} - α) + \cos^{2} γ = 1$ [given $α + β = \frac{π}{2}$ ]

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

$\Rightarrow 1 + \cos^{2} γ = 1$

$\Rightarrow \cos^{2} γ = 0$

$\Rightarrow \cos γ = 0$

$∴ {(\cos α + \cos β + \cos γ)}^{2} = {(\cos α + \sin α)}^{2}$

$= 1 + 2 \sin α \cdot \cos α = 1 + \sin 2 α$

Stream:BITSATSubject:MathematicsTopic:3D-Coordinate GeometrySubtopic:Coordinates of a point in space, Direction cosines,

⏱ 2mℹ️ Source: PYQ_2018

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