Step 1: Concept Definition:
The angles $\alpha, \beta, \gamma$ are direction angles for a line. Their cosines, $\cos\alpha, \cos\beta, \cos\gamma$, are the direction cosines. A fundamental identity connects these values.
Step 2: Core Formula:
The identity for direction cosines ($l, m, n$) is:
\[ l^2 + m^2 + n^2 = 1 \]
where $l = \cos\alpha$, $m = \cos\beta$, and $n = \cos\gamma$. This means $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$. The trigonometric Pythagorean identity is also relevant: $\sin^2\theta + \cos^2\theta = 1$.
Step 3: Derivation:
We aim to find $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$.
Using $\sin^2\theta = 1 - \cos^2\theta$, we get:
\[ \sin^2\alpha + \sin^2\beta + \sin^2\gamma = (1 - \cos^2\alpha) + (1 - \cos^2\beta) + (1 - \cos^2\gamma) \]
Rearranging:
\[ = (1 + 1 + 1) - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) \]
\[ = 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) \]
Substituting the fundamental identity $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$:
\[ = 3 - 1 = 2 \]
Step 4: Conclusion:
The value of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is 2.