First slide

Direction ratios and direction cosines

Question

If a line makes angles $α, β, γ$ with coordinate axes respectively, then the value of $\frac{\sum \cos 2 α}{\sum \cos^{2} α + \sum \sin^{2} α} =$

Moderate

A

$\frac{1}{2}$
B

$\frac{3}{2}$
C

$- \frac{1}{2}$
D

$- \frac{1}{3}$

Solution

If $α, β, γ$ are the angles made by a line with axes in the positive direction then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$ and $\sin^{2} α + \sin^{2} β + \sin^{2} γ = 1$
And consider the expression $\cos 2 α + \cos 2 β + \cos 2 γ$
Use the trigonometry identity $\cos 2 α = 2 \cos^{2} α - 1$
It implies that $\begin{matrix} \cos 2 α + \cos 2 β + \cos 2 γ = 2 (\cos^{2} α + \cos^{2} β + \cos^{2} γ) - 3 \\ = - 1 \end{matrix}$
Therefore, $\frac{\sum \cos 2 α}{\sum \cos^{2} α + \sum \sin^{2} α} = \frac{- 1}{1 + 2} = - \frac{1}{3}$

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