Problem 48 - A Complete Resource Book in Mathematics for JEE Main

Question

A line makes the same angle \(\theta\), with each of the \(x\) and \(z\) axis. If the angle \(\beta\), which it makes with \(y\)-axis, is such that \(\sin ^{2} \beta=3 \sin ^{2} \theta\), then \(\cos ^{2} \theta\) equals (A) \(\frac{2}{3}\) (B) \(\frac{1}{5}\) (C) \(\frac{3}{5}\) (D) \(\frac{2}{5}\)

Step-by-Step Solution

Answer

\( \cos^2 \theta = \frac{3}{5} \), thus the answer is (C).

Step 1: Understanding Direction Cosines

The direction cosines of a line are the cosines of the angles that the line makes with the coordinate axes. If a line makes angles \( \theta, \beta, \gamma \) with the \( x, y, z \) axes respectively, then the direction cosines can be denoted by \( l = \cos \theta, m = \cos \beta, n = \cos \gamma \).

Step 2: Identify Equal Angles and Conditions

The line makes the same angle \( \theta \) with the \( x \) and \( z \) axes. So, \( \cos \theta \) is the direction cosine for both axes: \( l = n = \cos \theta \). We're also given that \( \sin^2 \beta = 3 \sin^2 \theta \).

Step 3: Relation of Direction Cosines

Using the identity for a line in a 3D space, the sum of squares of direction cosines is 1: \( l^2 + m^2 + n^2 = 1 \). Substituting the known values, \( \cos^2 \theta + \cos^2 \beta + \cos^2 \theta = 1 \).

Step 4: Express \( \cos^2 \beta \) in terms of \( \sin^2 \beta \)

We have \( \sin^2 \beta = 3 \sin^2 \theta \). Therefore, \( \cos^2 \beta = 1 - \sin^2 \beta = 1 - 3 \sin^2 \theta \).

Step 5: Substitute \( \cos^2 \theta \) and \( \cos^2 \beta \) in identity and Simplify

From Step 3, we have \( 2 \cos^2 \theta + (1 - 3 \sin^2 \theta) = 1 \). Since \( \cos^2 \theta = 1 - \sin^2 \theta \), substituting into the equation gives \( 2(1 - \sin^2 \theta) + (1 - 3 \sin^2 \theta) = 1 \).

Step 6: Solve for \( \sin^2 \theta \)

Simplifying the equation, get \( 2 - 2\sin^2 \theta + 1 - 3\sin^2 \theta = 1 \). Combine \( -2\sin^2 \theta - 3\sin^2 \theta = -5\sin^2 \theta \) and solve \( 3 = 1 + 5\sin^2 \theta \) to find \( \sin^2 \theta = \frac{2}{5} \).

Step 7: Calculate \( \cos^2 \theta \) and Conclusion

Knowing \( \sin^2 \theta = \frac{2}{5} \), find \( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{2}{5} = \frac{3}{5} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that involve functions such as sine, cosine, and tangent. These identities are crucial tools in geometry and trigonometry, helping to solve problems and simplify expressions.

Some essential trigonometric identities include:

• Pythagorean Identity: \ \( \sin^2 \theta + \cos^2 \theta = 1 \ \)
• Double Angle Identity: \ \( \sin 2\theta = 2 \sin \theta \cos \theta \ \)
• Complementary Angle Identity: \ \( \sin (90^\circ - \theta) = \cos \theta \ \)

In problems involving direction cosines in 3D geometry, such as this one, the Pythagorean identity plays a key role. Specifically, it helps relate the square of the sine and cosine of angles. In this exercise, we substitute and use identity transformations to express the square of sine in terms of cosine and vice-versa, ultimately solving for \( \cos^2 \theta \).

Understanding these identities can make it easier to manipulate expressions and solve trigonometric equations quickly.

3D Geometry

3D geometry extends the principles of plane geometry into the third dimension, bringing in concepts that deal with points, lines, planes, and figures in space. This subject is pivotal when understanding how lines and planes interact with each other.

The core elements of 3D geometry include:

• Coordinate Axes: The x, y, and z axes form the basis of the Cartesian coordinate system in three dimensions.
• Points and Lines: A point in 3D is described by coordinates (x, y, z) while a line usually is represented in parametric or vector form.
• Planes: Defined by an equation \( ax + by + cz = d \).
• Direction Cosines: The cosines of the angles that a line makes with the x, y, and z axes are called the direction cosines.

In this problem, we deal with the direction cosines of a line in 3D. We are given their values concerning angles formed between the line and the coordinate axes, specifically those with the x and z axes, both being \( \theta \), while the one with the y axis is \( \beta \). The task focuses on utilizing these concepts to find precise trigonometric values.

Angle with Axes

In three-dimensional space, a line can make angles with each of the coordinate axes, namely the x, y, and z axes. These angles are crucial as they define the orientation of the line in space.

The angles that a line makes with the axes are often denoted as \( \theta \), \( \beta \), and \( \gamma \) for the x, y, and z axes, respectively. The cosines of these angles are known as the direction cosines.

• These are given as \( l = \cos \theta \), \( m = \cos \beta \), \( n = \cos \gamma \), which follow the relationship: \ \( l^2 + m^2 + n^2 = 1 \ \)

In this exercise, the line makes the same angle with the x and z axes (\( \theta \)), simplifying orientation calculations. Meanwhile, the problem tells us that \( \sin ^{2} \beta = 3 \sin ^{2} \theta \), which links the angle with the y-axis in relation to those with the other axes. Understanding how these angles interplay allows us to find out unknown angles or, in this case, trigonometric identities of a known angle like \( \cos^2 \theta \). Mastering this concept offers deeper insight into analyzing lines in 3D space.