Question

If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

• A. 1
• B. 2
• C. 3
• D. -2

Hint:

This is a standard identity in 3D geometry. It's useful to memorize both forms: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ and $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$.

Answer 1

The Correct Option is B

Step 1: Concept Introduction:
The angles $\alpha, \beta, \gamma$ represent the direction angles of a line. The cosines of these angles, $\cos\alpha, \cos\beta, \cos\gamma$, are known as the direction cosines. A key relationship exists between these direction cosines.Step 2: Core Principle:
The fundamental identity for direction cosines ($l, m, n$) states:
\[ l^2 + m^2 + n^2 = 1 \]Here, $l = \cos\alpha$, $m = \cos\beta$, and $n = \cos\gamma$. Consequently, $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$. We also utilize the trigonometric Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$.
Step 3: Derivation:
We aim to determine the value of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$.
Applying the identity $\sin^2\theta = 1 - \cos^2\theta$, the expression transforms to:
\[ \sin^2\alpha + \sin^2\beta + \sin^2\gamma = (1 - \cos^2\alpha) + (1 - \cos^2\beta) + (1 - \cos^2\gamma) \]Grouping terms yields:
\[ = (1 + 1 + 1) - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) \]\[ = 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) \]Substituting the fundamental identity $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$:
\[ = 3 - 1 = 2 \]Step 4: Conclusion:
The computed value for $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is 2.