Question

Each of the angles \( \beta \) and \( \gamma \) that a given line makes with the positive y- and z-axes, respectively, is half the angle that this line makes with the positive x-axis. Then the sum of all possible values of the angle \( \beta \) is

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

Hint:

Use the relationship between the direction cosines of a line and the angles it makes with the coordinate axes: \( l = \cos \alpha \), \( m = \cos \beta \), \( n = \cos \gamma \), and \( l^2 + m^2 + n^2 = 1 \). Substitute the given relationships between the angles and solve the resulting trigonometric equation. Remember to consider the possible range of angles with the positive axes.

Answer 1

The Correct Option is A

To solve the problem, let's consider the given conditions regarding the angles a line makes with the axes. Denoting large capital angles: \( \alpha \) with the x-axis, \( \beta \) with the y-axis, and \( \gamma \) with the z-axis.

According to the problem statement:

• \( \beta = \frac{\alpha}{2} \)
• \( \gamma = \frac{\alpha}{2} \)
• We need to find the sum of all possible values of the angle \( \beta \).

The direction cosines of the line with respect to the x-, y-, and z-axes are given by:

• \(\cos \alpha\), \(\cos \beta\), \(\cos \gamma\)

According to the identity involving direction cosines, we have:

\(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\)

Substitute the known relations into the equation:

• \(\cos^2 \alpha + \left(\cos \frac{\alpha}{2}\right)^2 + \left(\cos \frac{\alpha}{2}\right)^2 = 1\)

This simplifies to:

\(\cos^2 \alpha + 2 \cos^2 \frac{\alpha}{2} = 1\)

Using the double angle identity: \( \cos^2 \frac{\alpha}{2} = \frac{1 + \cos \alpha}{2} \), we substitute and get:

\(\cos^2 \alpha + 2 \times \frac{1 + \cos \alpha}{2} = 1\)

This further simplifies to:

\(\cos^2 \alpha + 1 + \cos \alpha = 1\)

Or, reorganizing terms:

\(\cos^2 \alpha + \cos \alpha = 0\)

Factor the quadratic equation:

\(\cos \alpha (\cos \alpha + 1) = 0\)

This gives us two solutions:

• \( \cos \alpha = 0 \Rightarrow \alpha = \frac{\pi}{2} \)
• \( \cos \alpha = -1 \Rightarrow \alpha = \pi \)

Substituting back, we find the possible values of \( \beta \):

• For \(\alpha = \frac{\pi}{2}\):
  • \(\beta = \frac{\pi}{4}\)
• For \(\alpha = \pi\):
  • \(\beta = \frac{\pi}{2}\)

The sum of all possible values of \( \beta \) is then:

\(\frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4}\)

Therefore, the sum of all possible values of the angle \( \beta \) is \(\frac{3\pi}{4}\).

Answer 2

Let the angle with the positive x-axis be \( \alpha \).

\(\text{Given, } \beta = \frac{\alpha}{2} \text{ and } \gamma = \frac{\alpha}{2}. \)

\(\text{We know that } \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1. \)

\(\text{Substituting the values of } \beta \text{ and } \gamma: \)

\(\cos^2 \alpha + \cos^2 \left( \frac{\alpha}{2} \right) + \cos^2 \left( \frac{\alpha}{2} \right) = 1 \)

\(\Rightarrow \cos^2 \alpha + 2 \cos^2 \left( \frac{\alpha}{2} \right) = 1 \)

\(\text{Using the identity } \cos \alpha = 2 \cos^2 \left( \frac{\alpha}{2} \right) - 1, \text{ we get } \)

\(2 \cos^2 \left( \frac{\alpha}{2} \right) = \cos \alpha + 1 \)

\(\text{So,} \)

\(\cos^2 \alpha + \cos \alpha + 1 = 1 \)

\(\Rightarrow \cos^2 \alpha + \cos \alpha = 0 \)

\(\Rightarrow \cos \alpha (\cos \alpha + 1) = 0 \)

\(\text{This gives } \cos \alpha = 0 \text{ or } \cos \alpha = -1. \)

Case 1:

\(\cos \alpha = 0 \)

\(\Rightarrow \alpha = \frac{\pi}{2} \text{ or } \alpha = \frac{3\pi}{2} \)

\(\text{Since the angles are with the positive axes, } 0 \le \alpha, \beta, \gamma \le \pi. \)

\(\text{If } \alpha = \frac{\pi}{2}, \text{ then } \beta = \frac{\pi}{4} \)

\(\text{If } \alpha = \frac{3\pi}{2}, \text{ this is not possible as } \beta = \frac{3\pi}{4} \text{ and } \gamma = \frac{3\pi}{4}, \)

\(\text{leading to } \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 0 + \frac{1}{2} + \frac{1}{2} = 1. \)

Case 2:

\(\cos \alpha = -1 \)

\(\Rightarrow \alpha = \pi \)

\(\Rightarrow \beta = \frac{\pi}{2}, \quad \gamma = \frac{\pi}{2} \)

\(\Rightarrow \cos^2 \pi + \cos^2 \frac{\pi}{2} + \cos^2 \frac{\pi}{2} = 1 + 0 + 0 = 1 \)

\(\text{Possible values of } \beta \text{ are } \frac{\pi}{4} \text{ and } \frac{\pi}{2}. \)

\(\text{Sum of possible values of } \beta = \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4}. \)