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In three-dimensional geometry, any line can be described by the angles it forms with the coordinate axes, namely, the x-axis, y-axis, and z-axis. These angles help us determine the line's direction cosines, which are essentially the cosine values of these angles.
Direction cosines are fundamental in understanding how a line is oriented in three-dimensional space. For a line making angles \(\theta_x\), \(\theta_y\), and \(\theta_z\) with the x-axis, y-axis, and z-axis respectively, their direction cosines are \(\cos \theta_x\), \(\cos \theta_y\), and \(\cos \theta_z\).

• The sum of the squares of the direction cosines equals 1, providing a critical relationship: \[ \cos^2 \theta_x + \cos^2 \theta_y + \cos^2 \theta_z = 1 \]

This equation helps ensure the line maintains its orientation throughout its length, and it is used ubiquitously in vector and analytical geometry to describe lines in a 3D space.

The angles that a line makes with the coordinate axes, denoted here as \( \theta \) for the x and z axes and \( \beta \) for the y-axis, are crucial for understanding a line's orientation. In the given problem, these angles are utilized to solve specific geometric relationships.
When a line makes equal angles with two axes, it implies symmetrical alignment in space, providing specific insights into the geometric properties of the line. The given problem takes advantage of this symmetry by assuming \( \theta_x = \theta_z = \theta \), simplifying calculations significantly.

• This symmetry leads directly to using key formulas related to the direction cosines in the problem solution.
• The relationship between these angles and the trigonometric function \( \sin \) helps translate one angle into another.

Using these known angles, we can solve the problems by using trigonometric identities, making it much easier to derive further geometric properties and equations.

Trigonometric identities form the backbone of many solutions in geometry, especially in problems involving angles and orientations. Here, they play a role in expressing relationships between angles \( \theta \) and \( \beta \).
One useful identity is \( \sin^2 \theta + \cos^2 \theta = 1 \), which allows conversion between sine and cosine functions. In this problem, rather than looking purely at cosines, the given condition \( \sin^2 \beta = 3 \sin^2 \theta \) translates into another form through this identity.

• By rewriting \( \sin^2 \beta = 1 - \cos^2 \beta \) and substituting, we find that \( \cos^2 \beta = 3 \cos^2 \theta - 2 \).
• This conversion step is critical, allowing for the existing equation \( 2\cos^2 \theta + \cos^2 \beta = 1 \) to simplify and solve the problem.

Understanding how to apply and manipulate these identities properly is central to succeeding in such exercises, as it allows for simplification and solving of otherwise complex-looking mathematical relationships.