First slide

Direction ratios and direction cosines

Question

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

Easy

A

$1 + \sin 2 β$
B

$1 - \sin 2 β$
C

$1 - \cos 2 β$
D

$\cos^{2} α$

Solution

If $α, β, γ$ are the angles made by a line ‘ $L$ ’ with the coordinate axes in the positive direction then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
Given $α, β, γ = \frac{π}{2} - β$
It implies that $\cos^{2} α + \cos^{2} β + \sin^{2} β = 1 \Rightarrow \cos^{2} α = 0$
Consider the expression
$\begin{matrix} {(\cos α + \cos β + \cos γ)}^{2} = {(0 + \cos β + \sin β)}^{2} \\ = \cos^{2} β + \sin^{2} β + 2 \cos β \sin β \\ = 1 + \sin 2 β \end{matrix}$
Therefore, the required expression is $1 + \sin 2 β$

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