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A line makes the same angle $\Theta$ , with each of the $x\; and\; z\;$ axis. If the angle $\beta$ , which it makes with $y$ -axis, is such that $\sin ^{2}\beta =3\sin ^{2}\Theta ,then\; \cos ^{2}\Theta$ equals:

Option 1)
3/5

Option 2)
1/5

Option 3)
2/3

Option 4)
2/5

Answers (1)

best_answer

As we learnt in

Direction Cosines -

If $\alpha, \beta, \gamma$ are the angles which a vector makes with positive X-axis,Y-axis and Z-axis respectively then $\cos \alpha, \cos \beta, \cos \gamma$ are known as diresction cosines, generally denoted by (l,m,n).

$l=\cos \alpha, m=\cos \beta, n=\cos \gamma$

$l^{2}+m^{2}+n^{2}= 1$

$\cos^{2} \alpha+ \cos^{2} \beta+\cos^{2} \gamma= 1$

- wherein

Given $\alpha =\gamma= \theta$

Also, $\sin ^{2}\beta = 3\sin ^{2}\theta$

$1- \cos ^{2}\beta = 3- 3\cos ^{2}\theta$

$\cos ^{2}\beta = 3\cos ^{2}\theta-2$

$\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma = 1$

$5\cos ^{2}\theta -2 = 1$

$\cos ^{2}\theta= \frac{3}{5}$

Option 1)

3/5

Correct option

Option 2)

1/5

Incorrect option

Option 3)

2/3

Incorrect option

Option 4)

2/5

Incorrect option

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prateek

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