Problem 94 - 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. $$\left|\begin{array}{ccc} \cos \theta & -r \sin \theta & 0 \\ \sin \theta & r \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is \(r\).

Step 1: Recall the formula for a 3x3 determinant

To find the determinant of a 3x3 matrix, use the formula: \[\text{det}(A) = a(ei−fh)−b(di−fg)+c(dh−eg)\]where the matrix \[A = \begin{pmatrix}a & b & c \d & e & f \g & h & i \end{pmatrix}\]The terms \(a, b, c, d, e, f, g, h, i\) are elements of the matrix.

Step 2: Identify the elements from the given matrix

Given the matrix: \[\begin{pmatrix}\cos \theta & -r \sin \theta & 0 \\sin \theta & r \cos \theta & 0 \0 & 0 & 1\end{pmatrix}\]These are the respective elements: \(a = \cos \theta\), \(b = -r \sin \theta\), \(c = 0\), \(d = \sin \theta\), \(e = r \cos \theta\), \(f = 0\), \(g = 0\), \(h = 0\), \(i = 1\).

Step 3: Apply the formula for the determinant

Substitute the identified elements into the determinant formula:\[\text{det}(A) = \cos \theta ((r \cos \theta)(1) - (0)(0)) - (-r \sin \theta)((\sin \theta)(1) - (0)(0)) + 0 ((\sin \theta)(0) - (0)(r \cos \theta))\]This simplifies to:\[\text{det}(A) = \cos \theta (r \cos \theta) + r \sin \theta (\sin \theta) \]

Step 4: Simplify the expression

Simplify the expression obtained:\[\text{det}(A) = r \cos^2 \theta + r \sin^2 \theta \]Use the trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\). Hence, the expression simplifies to:\[\text{det}(A) = r \times 1 = r\]

Step 5: Conclude the determinant

The determinant of the given 3x3 matrix simplifies to \(r\).

Key Concepts

3x3 Matrix Determinant

Determinants are a fascinating part of linear algebra that extend into calculus, especially in calculating integrals involving multiple variables. The determinant of a 3x3 matrix offers a single numerical value that can provide insights about the matrix. For example, in calculus, determinants can help in evaluating double and triple integrals when changing variables. Consider the given 3x3 matrix:

• The top row has elements: \( \cos \theta, -r \sin \theta, 0 \)
• The middle row has elements: \( \sin \theta, r \cos \theta, 0 \)
• The bottom row has elements: \( 0, 0, 1 \)

To calculate the determinant, apply the standard formula: \[\text{det}(A) = a(ei−fh)−b(di−fg)+c(dh−eg)\]For the given matrix, this requires replacing each letter with the corresponding element from the matrix and simplifying. The third column contributes no value due to zero elements, making our calculation simpler. In summary, each section of the determinant represents the matrix's impact on space, which, in turn, relates to the integrals.

Trigonometric Functions

Trigonometric functions like sine and cosine play crucial roles both in solving the determinant and their application in calculus. These trigonometric expressions help modify the coordinates or functions of variables in similar ways that deal with angular relationships and periodic motions. In the given matrix, trigonometric functions appear in the first and second rows:

• \( \cos \theta \) in the first row
• \( \sin \theta \) in the second row

They support transforming the coordinate system. Moreover, the trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \) simplifies complex expressions. This specific identity allowed us to simplify the determinant calculation significantly, reducing the expression to an intuitive form, simply \( r \). Knowing these identities by heart can often make solving such problems more efficient.

Change of Variables in Integrals

The change of variables in integrals is an essential concept when tackling multivariable calculus problems, using determinants as a tool. The determinant value often appears in the Jacobian, a key component in changing variables for integration. When you change variables in an integral, say from \((x, y)\) to \((u, v)\), you use a transformation that involves the determinant of the Jacobian matrix:

• The Jacobian matrix is derived from partial derivatives of each new variable with respect to the old ones.
• In essence, its determinant acts as a scaling factor to adjust the volume element in integration.

For the given problem, these steps would apply when you change variables involving polar coordinates, where \( r \) becomes critical. It highlights how calculus and linear algebra intersect via the determinants, reinforcing that these elements enable accurate and properly-scaled integrations in transformed spaces.