First slide

Direction ratios and direction cosines

Question

Let a line makes an angle $θ$ with $y -, z -$ axes and $ϕ$ with $x -$ axis. If $\sin ϕ$ and $\sin θ$ are in the ratio $m : n$ , then $\cos θ =$

Moderate

A

$\frac{2 m}{2 n^{2} + m^{2}}$
B

$\frac{m}{2 n^{2} + m^{2}}$
C

$\frac{m}{\sqrt{2 n^{2} + m^{2}}}$
D

$\frac{n}{\sqrt{2 n^{2} + m^{2}}}$

Solution

Given $ϕ, θ, θ$ are the angles made by a line with axes, it implies that
$\begin{matrix} \cos^{2} ϕ + \cos^{2} θ + \cos^{2} θ = 1 \\ 2 \cos^{2} θ = 1 - \cos^{2} ϕ \\ = \sin^{2} ϕ \end{matrix}$
Given   $\frac{\sin ϕ}{\sin θ} = \frac{m}{n} \Rightarrow \sin ϕ = \frac{m}{n} \sin θ$
Substitute   $\sin ϕ = \frac{m}{n} \sin θ$ in the equation   $2 \cos^{2} θ = \sin^{2} ϕ$
It implies that
$\begin{matrix} 2 \cos^{2} θ = \frac{m^{2}}{n^{2}} \sin^{2} θ \\ \tan^{2} θ = \frac{2 n^{2}}{m^{2}} \end{matrix}$
Use right triangle, we get $\cos θ = \frac{m}{\sqrt{2 n^{2} + m^{2}}}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App