First slide

Direction ratios and direction cosines

Question

The direction angles made by a line ' $L$ ’ with the coordinate axes, are $α, β, γ$ . Then the range of $\sin^{2} α + \sin^{2} β$

Moderate

A

$ϕ$
B

$(0, 1)$
C

$[1, 2]$
D

$ℝ$

Solution

If $α, β, γ$ are the angles made by a line ‘L’ with the coordinate axes in the positive direction then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
It implies that $1 - \sin^{2} α + 1 - \sin^{2} β + 1 - \sin^{2} γ = 1 \Rightarrow \sin^{2} α + \sin^{2} β + \sin^{2} γ = 2$
Hence, $\sin^{2} α + \sin^{2} β = 2 - \sin^{2} γ$
The minimum value of $\sin^{2} γ$ is zero and its maximum value is 1
So that the range of $\sin^{2} α + \sin^{2} β$ is $[1, 2]$

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