Problem 91 - Precalculus

Question

In calculus, determinants are used when evaluating double and triple integrals through a change of variables. In these cases, the elements of the determinant are functions. Find each determinant. $$\left|\begin{array}{cc} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is \( r \).

Step 1: Write Down the Determinant Formula

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by: \( ad - bc \). We'll apply this formula to the provided matrix.

Step 2: Identify Matrix Elements

From the matrix \( \begin{pmatrix} \cos \theta & -r \sin \theta \ \sin \theta & r \cos \theta \end{pmatrix} \), identify the elements where \( a = \cos \theta \), \( b = -r \sin \theta \), \( c = \sin \theta \), and \( d = r \cos \theta \).

Step 3: Substitute Elements into Determinant Formula

Substitute the identified elements into the determinant formula: \( (\cos \theta)(r \cos \theta) - (-r \sin \theta)(\sin \theta) \).

Step 4: Simplify the Expression

Calculate each term separately: the first term is \( r \cos^2 \theta \) and the second term \( -(-r \sin \theta \sin \theta) = r \sin^2 \theta \).

Step 5: Combine and Further Simplify

Combine the terms \( r \cos^2 \theta + r \sin^2 \theta \). Use the Pythagorean trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to simplify this to \( r(\cos^2 \theta + \sin^2 \theta) = r(1) \).

Step 6: Write Down Final Simplified Result

The expression simplifies to \( r \). This is the value of the determinant.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices, rectangular arrays of numbers, symbols, or expressions. In calculus, determinants play a crucial role when solving problems involving these matrices. A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible (a non-zero determinant) or singular (a zero determinant).

For a 2x2 matrix, the determinant can be calculated using the formula:

• If the matrix is \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is \( ad - bc \).

This formula helps us find out how the matrix affects an area or volume as part of the transformation it represents.

When dealing with more complex matrices or higher dimensions, the determinant helps in describing how these transformations interact with multi-dimensional spaces. Understanding matrix algebra gives us the tools to manage and understand these transformations clearly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions like sine, cosine, and tangent that hold true for all values of the variables involved. They are essential tools in simplifying expressions, proving equations, and solving integrals in calculus.

One of the most important identities used in the context of determinants is the Pythagorean identity:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

This formula is particularly useful because it allows us to simplify expressions involving sine and cosine, like in the original step-by-step solution. By combining sine and cosine terms, we reduce their complexity significantly.

These identities act as shortcuts in algebraic processes allowing us to recognize patterns and quickly simplify or transform expressions, making complex problems much easier to solve. Mastery of these identities is crucial in understanding higher-level calculus concepts.

Change of Variables in Integration

The change of variables in integration is a method used to simplify complicated integrals by transforming them into more manageable forms. This powerful technique can drastically reduce the complexity involved in evaluating integrals over certain regions.

When we perform a change of variables, especially in multi-variable calculus, the Jacobian determinant is used. The determinant we've discussed earlier helps in adjusting the integral for this transformation. It accounts for the way regions of integration are stretched or compressed in the new coordinate system.

For example, if you're converting a double integral in polar coordinates, the determinant of the transformation matrix ensures that the area element is correctly adjusted according to the new variables. This makes the process smoother and helps in accurately evaluating the integral.

Understanding how to apply change of variables helps in visualizing integrals in different dimensions and making complex calculus problems much more approachable.