Question

If \(\alpha=46^{\circ}\) and \(\beta=108^{\circ}\) are direction angles for a vector \(\mathbf{u}\), find two possible values for the third angle.

Short answer

The two possible values for the third angle \( \gamma \) are approximately \( 49.5^{\circ} \) and \( 130.5^{\circ} \).

Step-by-step solution

Understand the Problem

Direction angles are angles that a vector makes with the positive axes (x, y, and z axes) in a three-dimensional space. The sum of the squares of the cosines of these angles equals 1.

Apply the Formula for Direction Angles

Use the equation for the direction angles of vector \( \mathbf{u} \): \[ \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1 \] where \( \alpha = 46^{\circ} \), \( \beta = 108^{\circ} \), and \( \gamma \) is the third direction angle.

Calculate the Cosine Values

Find \( \cos(\alpha) = \cos(46^{\circ}) \approx 0.6946 \) and \( \cos(\beta) = \cos(108^{\circ}) \approx -0.3090 \).

Substitute and Solve for \( \cos^2(\gamma) \)

Substitute the cosine values into the equation: \[ 0.6946^2 + (-0.3090)^2 + \cos^2(\gamma) = 1 \]Calculate the squares and sum: \[ 0.4824 + 0.0955 + \cos^2(\gamma) = 1 \]Solve for \( \cos^2(\gamma) \): \[ \cos^2(\gamma) = 1 - 0.5779 = 0.4221 \]

Determine Possible Values of \( \gamma \)

Since \( \cos^2(\gamma) = 0.4221 \), we find \( \cos(\gamma) = \pm \sqrt{0.4221} \).Calculate: \( \cos(\gamma) \approx \pm 0.6497 \).Finding \( \gamma \):1. \( \gamma \approx \cos^{-1}(0.6497) \approx 49.5^{\circ} \).2. \( \gamma \approx \cos^{-1}(-0.6497) \approx 130.5^{\circ} \).

Key concepts

Vectors in 3D Space

Imagine navigating through three-dimensional space. Vectors are essential tools in this realm, representing direction and magnitude. In 3D space, a vector can be visualized as an arrow pointing from one position to another. The vector can be defined by its components along the x, y, and z axes. These components describe how far the vector extends in each direction.

When working with vectors in 3D, we often refer to **direction angles**. These are the angles that a vector makes with the positive x, y, and z axes. They give us insight into the vector's orientation in space. To mathematically represent the contribution of each axis to the vector, the cosines of these direction angles are key. This is detailed in the equation:

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

where \(\alpha\), \(\beta\), and \(\gamma\) are direction angles. This relationship helps us ensure our vector is correctly oriented within 3D space.

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, play a crucial role in understanding angles and their relationships in geometry. The function \(\cos(\theta)\), where \(\theta\) is an angle, relates to the adjacent side of a right-angle triangle divided by the hypotenuse.

In the context of direction angles, the cosine function helps us quantify how much of a vector is aligned with each axis. Each angle corresponds to the cosine of that angle, representing how much the vector projects in the x, y, or z direction. For example, \( \cos(46^{\circ}) \approx 0.6946 \) indicates a relatively strong alignment with the axis, while \( \cos(108^{\circ}) \approx -0.3090 \) shows a weaker, but still significant, alignment.

Understanding these trigonometric functions is vital as they form the building blocks for solving problems related to vector orientation and manipulation.

Solving Trigonometric Equations

Solving trigonometric equations is an essential skill when dealing with direction angles. It involves manipulating equations to find unknown values, often using trigonometric identities or properties.

In the given exercise, you have the equation \[ \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1 \]. By substituting in the values of the known angles \(\alpha\) and \(\beta\), we first calculate their cosine values and square them. Post substitution and simplification, the equation allows us to isolate \( \cos^2(\gamma) \).

• Given \(\cos^2(\gamma) = 0.4221\), we take the square root to find \(\cos(\gamma) = \pm 0.6497 \).
• Using the inverse cosine function, we find that \(\gamma\) can take on different values of about \(49.5^{\circ}\) or \(130.5^{\circ}\) due to the periodic nature of cosine.

These steps illustrate the process of solving such trigonometric equations by using basic algebraic techniques in combination with trigonometric identities to find solutions to complex angle problems.