Question

If \(\alpha, \beta, \gamma\) are direction cosines of \(\underline{x}\), then \(\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=\) (A) 1 (B) 2 (C) \(-1\) (D) \(-2\)

Short answer

The value of the expression \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) is \(-1\) (Option C).

Step-by-step solution

Solving for \(\cos 2\alpha, \cos 2\beta, \cos 2\gamma\)

By applying the double angle identity, find the value of each \(\cos 2\alpha, \cos 2\beta, \cos 2\gamma\): \[\cos 2\alpha = 2\cos^2 \alpha - 1\] \[\cos 2\beta = 2\cos^2 \beta - 1\] \[\cos 2\gamma = 2\cos^2 \gamma - 1\]

Summing and Simplifying the cosines of double angles

Add the three equations and simplify: \[\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) - 3\] Now, recall the formula for the direction cosines: \[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\] Substitute this into the equation above and simplify: \[\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2(1) - 3\]

Solution

Finding the value of the expression: \[\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2 - 3 = -1\] Therefore, the value of \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) is \(-1\), which corresponds to answer choice (C).

Key concepts

Double Angle Identities

Understanding the double angle identities is essential for solving various trigonometric problems. These identities allow us to express the sine, cosine, or tangent of double angles, meaning angles of the form 2θ, in terms of the sine, cosine, or tangent of θ. For instance, the double angle identity for cosine is expressed as \[ \cos 2\alpha = 2\cos^2 \alpha - 1 \]. This particular form is useful for solving the above exercise as it links the cosine of the doubled angle to the square of the cosine of the original angle, which can then be easily related to direction cosines of a vector. It's also important to recognize that these identities can be rearranged to aid in simplication of equations, especially in exercises found in JEE mathematics and other advanced mathematics curriculums.

To master double angle identities, it's recommended that students practice regularly and understand how to derive them from the sum of angles formulas. This deepens comprehension and allows for application in a wide range of problems, especially those involving oscillatory movements or wave functions.

Vector Analysis in Mathematics

Vector analysis is a powerful tool used to describe and solve problems involving physical quantities that have both a magnitude and direction. In mathematics, vectors are often denoted in bold, such as \underline{x}, and can represent quantities like velocity, acceleration, or, in our example, direction in space. Direction cosines, referenced in our exercise, are a set of cosines of angles that a vector makes with the coordinate axes. They are denoted as \(\cos \alpha, \cos \beta, \cos \gamma\).

The three direction cosines of a vector are related by the identity \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\), which ensures that the vector's magnitude is preserved. This identity was used in our solution process and is indispensable in vector analysis problems. Understanding how to apply such vector identities within mathematical problems is an essential skill, especially within physics and engineering contexts.

JEE Mathematics Problems

The Joint Entrance Examination (JEE) is a challenging test for students in India aspiring to enter engineering programs. It covers an extensive mathematics syllabus, including algebra, calculus, and trigonometry. JEE Mathematics problems, like our example involving direction cosines, are designed to test the student's conceptual understanding, analytical thinking, and problem-solving abilities.

Success in JEE Mathematics requires not only memorizing formulas but also understanding their derivation and application in various scenarios. This includes exercises that integrate different mathematical concepts, such as trigonometric identities and vector analysis, to test the student’s ability to correlate and apply knowledge. Students preparing for JEE should focus on clarity of concepts, diligent practice, and developing a strategy to approach problems efficiently. Such skills are honed through solving a variety of problems, ranging from basic to complex, and learning from step-by-step solutions to grasp the underlying principles fully.