• # question_answer A line makes the same angle$\theta$with each of the x and z-axes. If the angle$\beta ,$which it makes with y-axis, is such that${{\sin }^{2}}\beta =3{{\sin }^{2}}\theta ,$then$\cos \theta$equals A) $\frac{2}{3}$   B)                                        $\frac{1}{5}$                    C) $\frac{3}{5}$                                    D) $\frac{2}{5}$

Correct Answer: C

Solution :

A line makes angles$\alpha ,\beta$respectively an$\gamma$with X-axis, Y-axis and Z-axis , then ${{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1.$ A line makes angle$\theta$with X-axis and Z-axis and $\beta$with Y-axis. $\therefore$$l=\cos \theta ,\text{ }m=\cos \beta ,\text{ }n=\cos \theta$ We know that, ${{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1$ $\Rightarrow$ ${{\cos }^{2}}\theta +{{\cos }^{2}}\beta +{{\cos }^{2}}\theta =1$ $\Rightarrow$          $2{{\cos }^{2}}\theta =1-{{\cos }^{2}}\beta$ $\Rightarrow$           $2{{\cos }^{2}}\theta ={{\sin }^{2}}\beta$                         ...(i) But            ${{\sin }^{2}}\beta =3\text{ }{{\sin }^{2}}\theta$                  ...(ii) From Eqs. (i) and (ii), we get $3{{\sin }^{2}}\theta =2{{\cos }^{2}}\theta$ $\Rightarrow$               $3(1-{{\cos }^{2}}\theta )=2{{\cos }^{2}}\theta$ $\Rightarrow$               $3=5{{\cos }^{2}}\theta$ $\Rightarrow$               ${{\cos }^{2}}\theta =\frac{3}{5}$

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