Question

If a line makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the positive directions of the x, y, and z-axis respectively, then \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) equals:

• A. 1
• B. -1
• C. 2
• D. -2
• E. 0

Hint:

Remember the double angle formulas and basic trigonometric identities when dealing with angle relationships in 3D geometry problems.

Answer 1

The Correct Option is B

From the spherical trigonometry, we know the identity for the sum of the squares of the direction cosines: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Using the double angle formula for cosine, \(\cos 2\theta = 2\cos^2 \theta - 1\), apply it to each angle: \[ \cos 2\alpha = 2\cos^2 \alpha - 1 \] \[ \cos 2\beta = 2\cos^2 \beta - 1 \] \[ \cos 2\gamma = 2\cos^2 \gamma - 1 \] Summing these expressions gives: \[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) - 3 \] Substitute the sum of the squares of the direction cosines: \[ = 2 \times 1 - 3 = 2 - 3 = -1 \]