If \( \alpha,\beta,\gamma \) are angles of inclinations of a line with \( x,y \) and \( z \) -axis respectively, then which of the following are correct?
A. \( \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 \)
B. \( \cos 2\alpha+\cos 2\beta+\cos 2\gamma=-1 \)
C. \( \sin^2\alpha+\sin^2\beta+\sin^2\gamma=-2 \)
D. \( \cos^2\alpha+\cos^2\beta+\cos^2\gamma=2 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A and B only
\( \cos\alpha,\cos\beta,\cos\gamma \) are the direction cosines of the line, so the sum of their squares is 1.
\( \displaystyle \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 \)
Using \( \cos 2\theta=2\cos^2\theta-1 \):
\( \displaystyle \cos 2\alpha+\cos 2\beta+\cos 2\gamma=2(\cos^2\alpha+\cos^2\beta+\cos^2\gamma)-3 \)
\( \displaystyle =2-3=-1 \)
Using \( \sin^2\theta=1-\cos^2\theta \):
\( \displaystyle \sin^2\alpha+\sin^2\beta+\sin^2\gamma=3-(\cos^2\alpha+\cos^2\beta+\cos^2\gamma) \)
\( \displaystyle =3-1=2 \)
So \( \sum\cos^2=1 \), \( \sum\cos 2\theta=-1 \) and \( \sum\sin^2=2 \).
Checking the statements
(A) \( \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 \) — Correct. This is the basic property of direction cosines.
(B) \( \cos 2\alpha+\cos 2\beta+\cos 2\gamma=-1 \) — Correct. \( 2(1)-3=-1 \).
(C) \( \sin^2\alpha+\sin^2\beta+\sin^2\gamma=-2 \) — Incorrect. The sum is \( 3-1=2 \); a sum of squares cannot be negative.
(D) \( \cos^2\alpha+\cos^2\beta+\cos^2\gamma=2 \) — Incorrect. The sum of the squares of the direction cosines is 1.