Question

If a straight line makes the angles \(60^\circ,45^\circ,\alpha\) with x, y and z axes respectively, then \( \sin^2\alpha \) is

• A. \( \frac{3}{4} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{1}{2} \)
• D. \( 1 \)
• E. \( \frac{1}{4} \)

Hint:

Whenever a line makes angles with axes, immediately apply the identity \( l^2 + m^2 + n^2 = 1 \). It simplifies the problem drastically.

Answer 1

The Correct Option is A

Concept: When a line makes angles \( \alpha, \beta, \gamma \) with the positive directions of x, y, and z axes respectively, then the cosines of these angles are called the direction cosines of the line. These satisfy the fundamental identity: \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \] This identity comes from the fact that direction cosines represent components of a unit vector in 3D space.

Step 1: Identify the given angles
The line makes angles: \[ 60^\circ \text{ with x-axis}, \quad 45^\circ \text{ with y-axis}, \quad \alpha \text{ with z-axis} \] So we can write: \[ \cos^2 60^\circ + \cos^2 45^\circ + \cos^2 \alpha = 1 \]

Step 2: Substitute known trigonometric values
We know standard values: \[ \cos 60^\circ = \frac{1}{2} \Rightarrow \cos^2 60^\circ = \frac{1}{4} \] \[ \cos 45^\circ = \frac{1}{\sqrt{2}} \Rightarrow \cos^2 45^\circ = \frac{1}{2} \] Substituting: \[ \frac{1}{4} + \frac{1}{2} + \cos^2 \alpha = 1 \]

Step 3: Simplify the equation carefully
First add the fractions: \[ \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \] So the equation becomes: \[ \frac{3}{4} + \cos^2 \alpha = 1 \]

Step 4: Solve for \( \cos^2 \alpha \)
\[ \cos^2 \alpha = 1 - \frac{3}{4} = \frac{1}{4} \]

Step 5: Convert to \( \sin^2 \alpha \)
Using identity: \[ \sin^2 \alpha = 1 - \cos^2 \alpha \] Substitute: \[ \sin^2 \alpha = 1 - \frac{1}{4} = \frac{3}{4} \]

Step 6: Interpretation and validation
Since direction cosines correspond to real geometric directions: - The value must lie between 0 and 1 - \( \frac{3}{4} \) satisfies this - Hence the result is consistent and valid

Step 7: Final Answer
\[ \boxed{\frac{3}{4}} \]