A straight line makes angles of 45° and 60° with the positive direction of the x and y-axes respectively.

Question

A straight line makes angles of \(45^{\circ}\) and \(60^{\circ}\) with the positive direction of the x and y-axes respectively. What angle does it make with the positive direction of the z-axis ? direction of the z-axis?

Answer 1

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

\( \cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} x=1\)

\(\left(\frac{1}{\sqrt{2}}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\cos ^{2} x=1\)

\(\cos ^{2} x=1-\frac{1}{2}-\frac{1}{4}=\frac{4-2-1}{4}=\frac{1}{4}\)

\(\cos x=\frac{1}{2} \)

\(\Rightarrow \cos x=\frac{-1}{2}\) because line is in 3rd quadrant

\(\therefore x = 60 ^\circ\)