Question

If a line makes angles of ${{30}^{\circ }}$ and ${{45}^{\circ }}$ with x- axis and y- axis, then the angle made by it with z- axis is
(a) ${{45}^{\circ }}$
(b) ${{60}^{\circ }}$
(c) ${{120}^{\circ }}$
(d) None of these

Answer 1

Hint: Given question is based on 3 -D Geometry. We know in 3D the relation between direction cosines of a line ${{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=1$, where $\alpha ,\beta and \gamma $ are the direction cosines of a line from x-axis, y-axis and z-axis respectively. We use this cosine equation to solve the question. By putting the values in the equation and solving it, we get our desirable answer.

Formula Used:
${{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=1$

Complete step by step Solution:
Given that line makes an angle of ${{30}^{\circ }}$and ${{45}^{\circ }}$with x- axis and y-axis
We have to find out the angle made by the line with the z- axis.
Let the angle between the line and the z – axis be x.
So, the three direction angles (angles to each of the positive coordinate axis ) of the line are 30, 45 and x degrees.
The direction cosines of the line are the cosines of the direction angles.
So, the sum of the squares of the direction cosines is 1.
Which means ${{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1$
That is ${{\cos }^{2}}(30)+{{\cos }^{2}}(45)+{{\cos }^{2}}(x)=1$
Now we put the value of cos${{30}^{\circ }}$ , cos ${{45}^{\circ }}$in the above equation, we get
${{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=1$${{\left( \dfrac{\sqrt{3}}{2} \right)}^{2}}+{{\left( \dfrac{1}{\sqrt{2}} \right)}^{2}}+{{\cos }^{2}}x=1$
By solving the equation, we get
              $\dfrac{3}{4}+\dfrac{1}{2}+{{\cos }^{2}}x=1$
Thus we find out the value of ${{\cos }^{2}}x$
             ${{\cos }^{2}}x=1-\dfrac{3}{4}-\dfrac{1}{2}$
             ${{\cos }^{2}}x=-\dfrac{1}{2}$
      Where ${{a}^{2}}\ge 0$, a $\in $ R
           ${{\cos }^{2}}x=-\dfrac{1}{2}$is not possible

Hence, the correct option is d.

Note: In this type of question, students made mistakes in putting the correct equation and correct trigonometric values in the question. Always remember that a negative angle is not possible. The angle must be a real number. By taking care of some of these things, we get our correct answer.