If the angles made by a straight line with th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We know that if $\alpha, \beta, \gamma$ are the angles made by a line with the coordinate axes, then the sum of the squares of their cosines is $1

Vedclass · Accepted Answer

$\frac{\pi}{2}$

Vedclass · Answer

(C) We know that if $\alpha, \beta, \gamma$ are the angles made by a line with the coordinate axes, then the sum of the squares of their cosines is $1$, i.e.,$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Given that the angles are $\alpha, \frac{\pi}{2}-\alpha, \beta$.Substituting these values into the identity, we get:$\cos^2 \alpha + \cos^2 \left(\frac{\pi}{2}-\alpha\right) + \cos^2 \beta = 1$Since $\cos \left(\frac{\pi}{2}-\alpha\right) = \sin \alpha$, the equation becomes:$\cos^2 \alpha + \sin^2 \alpha + \cos^2 \beta = 1$Using the identity $\cos^2 \alpha + \sin^2 \alpha = 1$, we have:$1 + \cos^2 \beta = 1$$\cos^2 \beta = 0$$\cos \beta = 0$Since $\cos \beta = 0$, we have $\beta = \frac{\pi}{2}$.

If the angles made by a straight line with the coordinate axes are $\alpha, \frac{\pi}{2}-\alpha, \beta$, then $\beta$ is equal to

Similar Questions

Find the direction cosines of a line which makes equal angles with the coordinate axes.

If a line makes angles $90^{\circ}, 135^{\circ},$ and $45^{\circ}$ with the $x, y,$ and $z$-axes respectively,find its direction cosines.

If the direction cosines of two lines satisfy the equations $2l+m-n=0$ and $l^2-2m^2+n^2=0$, and $\theta$ is the angle between the lines, then $\cos \theta=$

If a line makes angles of measure $\frac{\pi}{6}$ and $\frac{\pi}{3}$ with $X$ and $Y$ axes respectively,then the angle made by the line with $Z$ axis is

The direction cosines $l, m, n$ of two lines satisfy the equations $3l + m + 5n = 0$ and $6mn - 2nl + 5lm = 0$. If $\theta$ is the angle between these lines, then $|\cos \theta| = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module