Problem 183 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The matrix \(\boldsymbol{R}(t)\) is defined by \(R(t)=\left[\begin{array}{cc}\cos t & \sin t \\ -\sin t & \cos t\end{array}\right]\). Show that \(\boldsymbol{R}(\mathbf{s}) \boldsymbol{R}(\boldsymbol{t})=\boldsymbol{R}(\mathbf{s}+\boldsymbol{t})\).

Step-by-Step Solution

Answer

The matrix multiplication \(R(s) \cdot R(t)\) simplifies to \(R(s+t)\), hence showing the required result.

Step 1: Define The Matrices

Define two matrices \(R(s)\) and \(R(t)\) as stated in the exercise. Now, \(R(s)\) equals \(\begin{bmatrix} \cos s & \sin s \\ -\sin s & \cos s \end{bmatrix}\) and \(R(t)\) equals \(\begin{bmatrix} \cos t & \sin t \\ -\sin t & \cos t \end{bmatrix}\).

Step 2: Multiply the Matrices

Multiply these matrices just as you would multiply any other matrices: row by column. The (i,j) entry of the resulting matrix is found by multiplying elements of the i-th row of the first matrix by corresponding elements of the j-th column of the second matrix, then summing those products.

Step 3: Simplify using Trigonometric Identities

While simplifying, you will encounter terms like \(\cos(s) \cdot \cos(t) + \sin(s) \cdot \sin(t)\) and \(\cos(s) \cdot \sin(t) - \sin(s) \cdot \cos(t)\). Use the cosine and sine addition identities, which are \(\cos(x+y) = \cos(x) \cdot \cos(y) - \sin(x) \cdot \sin(y)\) and \(\sin(x+y) = \sin(x) \cdot \cos(y) + \cos(x) \cdot \sin(y)\) respectively. This will simplify the resulting matrix which will look like the \(R(s+t)\) defined above.

Step 4: Write the Final Result

Upon simplification, the resulting matrix will look exactly like \(R(s+t)\), which completes the exercise.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions like sine and cosine, which are true for all the values of the variable(s) involved. These identities are very helpful in simplifying and solving trigonometric problems.

Some of the commonly used trigonometric identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Angle Sum Identities: - Cosine Addition Identity: \( \cos(x + y) = \cos x \cos y - \sin x \sin y \)
  - Sine Addition Identity: \( \sin(x + y) = \sin x \cos y + \cos x \sin y \)
• Double Angle Formulas: - \( \cos(2x) = \cos^2 x - \sin^2 x \)

These identities are not just random facts. They are derived through geometric interpretations or algebraic manipulations and used in mathematical calculations as tools to simplify and solve more complex trigonometric problems.

Rotation Matrices

Rotation matrices are used to rotate points in a plane or in space. They are a fundamental tool in linear algebra, especially within the field of computer graphics and robotics.

Consider the 2D rotation matrix:

• \( R(\theta) = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix} \)

This matrix rotates a point around the origin of the coordinate system by an angle \( \theta \) counterclockwise. The columns of the matrix represent the rotated axes.

The important feature of rotation matrices is that they preserve the length of vectors and the angle between vectors. Thus, they are a form of orthogonal matrix, ensuring that the transformation is purely a rotation, with no scaling or shearing. Using matrices for rotation is powerful because they can be easily composed by multiplying them together—offering an easy way to combine multiple rotations.

Cosine and Sine Addition Formulas

The cosine and sine addition formulas are specific trigonometric identities that help in finding the cosine and sine of the sum of two angles. These formulas break down a complex expression into simpler components that can be easily evaluated.

The formulas are:

• Cosine Addition Formula:
  \(\cos(x + y) = \cos(x) \cdot \cos(y) - \sin(x) \cdot \sin(y)\)
• Sine Addition Formula:
  \(\sin(x + y) = \sin(x) \cdot \cos(y) + \cos(x) \cdot \sin(y)\)

These identities are extremely useful in problems involving the sum and difference of angles, such as those found in physics, engineering, and computer graphics.

In the context of the original exercise, when multiplying two rotation matrices, the cosine and sine addition formulas allow for the simplification of terms in the resulting matrix. This ultimately shows that multiplying two rotation matrices is equivalent to a single rotation by the sum of the angles—demonstrating the practical applicability of these formulas.