Problem 246 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\left[\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]\left[\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right]^{-1}=\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]\), then show that \(a=\cos 2 \theta, b=\sin 2 \theta .\)

Step-by-Step Solution

Answer

It's been shown that \( a = \cos2\theta \) and \( b = \sin2\theta \).

Step 1: Computing the Multiplication of two Matrices

The multiplication of two matrices is performed element by element. Thus, to multiply out the matrices, multiply the elements of the first row of the first matrix by the corresponding elements of the first column of the second matrix and add the products. Repeat this process for all rows of the first matrix and all columns of the second matrix until every element of the new matrix has been calculated.

Step 2: The resulting matrix

After performing the multiplication, we obtain a new matrix \[ \left[ \begin{array}{cc} 1 + \tan^2(\theta) & 0 \\ 0 & 1 + \tan^2(\theta) \end{array} \right] \]. This is derived from the Pythagorean identity (1 + \tan^2(\theta) = \sec^2(\theta)), which allows us to simplify this to \[ \left[ \begin{array}{cc} \sec^2(\theta) & 0 \\ 0 & \sec^2(\theta) \end{array} \right] \]. This is the matrix that we are equating to \[ \left[ \begin{array}{cc} a & -b \\ b & a \end{array} \right] \].

Step 3: Equating the matrices

Equating the value of 'a' from the calculated matrix and the given matrix we get, \( \sec^2(\theta) = a \. Also, we get that 'b' equals zero. However, we know \( \sec^2(\theta) \) is equal to \( 1 + \tan^2(\theta) \), and using the double-angle formula, we can say that \( 1 + \tan^2(\theta) = 1 + 2\sin^2(\theta) = \cos2\theta \). Therefore \( a = \cos2\theta \). Since \( b = 0 \), this means \( b = \sin2\theta \) because \( \sin \) of any angle equals 0 if that angle is an integer multiple of \( \pi \).

Step 4: Final Answer

From the comparison of the calculated matrix with the given matrix, it is confirmed that \( a = \cos2\theta \) and \( b = \sin2\theta \). These are the final answers.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is fundamental in geometry and has applications across various fields, including physics, engineering, and even music. In trigonometry, angles are typically measured in degrees or radians, and several functions describe the relationships between these angles. The primary trigonometric functions are:

• Sine (\( heta\)) - Represents the ratio of the length of the side opposite to the angle to the hypotenuse of the triangle.
• Cosine (\( heta\)) - Represents the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\( heta\)) - Expresses the ratio of the sine to the cosine of an angle, thus opposing side length to adjacent side length.

Understanding these functions and their relationships is crucial for solving geometric problems and is especially relevant when manipulating matrices involving angles as shown in the exercise. By applying trigonometric identities and formulas, we can simplify and solve for unknowns like those found in matrix elements.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity derived from the Pythagorean theorem. It states that for any angle \( heta\),\( \sin^2( heta) + \cos^2( heta) = 1 \). This identity is crucial because it relates the sine and cosine functions. It allows us to express one of these functions in terms of the other, simplifying calculations.In the context of the exercise, the Pythagorean identity is adapted for the tangent function. We know that:

• \( \tan( heta) = \frac{\sin(\theta)}{\cos(\theta)}\)
• By using \(\tan^2(\theta) + 1 = \sec^2(\theta)\), we can further relate these functions.

Here, the Pythagorean identity helps simplify the matrix multiplication process and equates the resulting elements to \(a\), aiding in demonstrating that \(a = \cos(2\theta)\).

Double-angle Formulas

The double-angle formulas are trigonometric identities used to express trigonometric functions of double angles, usually written as \(2 heta\). These formulas are particularly useful in simplifying expressions and solving equations involving trigonometric functions. The key double-angle formulas are:

• For cosine: \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\) or equivalently \(\cos(2\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)\).
• For sine: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

In our exercise, these double-angle identities allow us to express the elements of the product matrix in terms of explicit trigonometric values. The substitution of these values lets us confidently affirm that \(a = \cos(2\theta)\) and \(b = \sin(2\theta)\). Understanding and applying these formulas directly helps unravel the complexities within trigonometric problems, especially in multi-step mathematical operations like matrix equations.