Question

If $\alpha$, $\beta$, $\gamma$ are direction angles of a line and $\alpha = 60^\circ$, $\beta = 45^\circ$, $\gamma =$ ____ .

• A. 30° and 90°
• B. 45° and 60°
• C. 90° and 30°
• D. 60° and 120°

Answer 1

Answer: (D)

60° and 120°

Answer 2

Direction angles $\alpha$, $\beta$, $\gamma$ of a line satisfy the fundamental relation: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

Given $\alpha = 60^\circ$ and $\beta = 45^\circ$. We know the values of $\cos(60^\circ)$ and $\cos(45^\circ)$:

$\cos(60^\circ) = \frac{1}{2}$ $\cos(45^\circ) = \frac{1}{\sqrt{2}}$

Substitute these values into the relation:

$\left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2\gamma = 1$ $\frac{1}{4} + \frac{1}{2} + \cos^2\gamma = 1$

Combine the fractions:

$\frac{1}{4} + \frac{2}{4} + \cos^2\gamma = 1$ $\frac{3}{4} + \cos^2\gamma = 1$

Now, solve for $\cos^2\gamma$:

$\cos^2\gamma = 1 - \frac{3}{4}$ $\cos^2\gamma = \frac{1}{4}$

Take the square root of both sides to find $\cos\gamma$:

$\cos\gamma = \pm\sqrt{\frac{1}{4}}$ $\cos\gamma = \pm\frac{1}{2}$

This gives two possible values for $\cos\gamma$:

1. If $\cos\gamma = \frac{1}{2}$, then $\gamma = 60^\circ$ (since direction angles are usually taken in the range $[0^\circ, 180^\circ]$).
2. If $\cos\gamma = -\frac{1}{2}$, then $\gamma = 120^\circ$ (since direction angles are usually taken in the range $[0^\circ, 180^\circ]$).

Thus, the possible values for $\gamma$ are $60^\circ$ and $120^\circ$.