Question

\(\left|\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right| \quad\left|\begin{array}{cc}\cos ^{2} \Phi & \cos \phi \sin \phi \\ \cos \phi \sin \phi & \sin ^{2} \Phi\end{array}\right|=\left|\begin{array}{cc}0 & 0 \\\ 0 & 0\end{array}\right|\) provided \(\theta-\Phi=\ldots . \mathrm{n} \in \mathrm{Z}\) (a) \(n \pi\) (b) \((2 \mathrm{n}+1)(\pi / 2)\) (c) \(\mathrm{n}(\pi / 2)\) (d) \(2 \mathrm{n} \pi\)

Short answer

The short answer to the given question is: \(\theta - \Phi = (2n + 1)(\pi / 2)\), hence the correct option is \(\boxed{\text{(b)}}\).

Step-by-step solution

Multiply the matrices

First, let's multiply the given matrices: \(\left|\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\\ \cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right| \quad\left|\begin{array}{cc}\cos ^{2} \Phi & \cos \phi \sin \phi \\\ \cos \phi \sin \phi & \sin ^{2} \Phi\end{array}\right|\) Recall the matrix multiplication formula, where the element in row i and column j of the product is the sum of the product of the ith row of the first matrix and the jth column of the second matrix. Using this, we get: \(\left|\begin{array}{cc} \cos^2{\theta}\cos^2{\Phi} + \cos{\theta}\sin{\theta}\cos{\phi}\sin{\phi} & \cos^2{\theta}\cos{\phi}\sin{\phi} + \cos{\theta}\sin{\theta}\sin^2{\Phi} \\ \cos{\theta}\sin{\theta}\cos^2{\Phi} + \sin^2{\theta}\cos{\phi}\sin{\phi} & \cos{\theta}\sin{\theta}\cos{\phi}\sin{\phi} + \sin^2{\theta}\sin^2{\Phi} \end{array}\right|\)

Equate the matrices to given condition and solve

Now we are given the condition that the product of these matrices is equal to: \(\left|\begin{array}{cc}0 & 0 \\\ 0 & 0\end{array}\right|\) Equating the product matrix we calculated in step 1, we get: \(\left|\begin{array}{cc} \cos^2{\theta}\cos^2{\Phi} + \cos{\theta}\sin{\theta}\cos{\phi}\sin{\phi} & \cos^2{\theta}\cos{\phi}\sin{\phi} + \cos{\theta}\sin{\theta}\sin^2{\Phi} \\ \cos{\theta}\sin{\theta}\cos^2{\Phi} + \sin^2{\theta}\cos{\phi}\sin{\phi} & \cos{\theta}\sin{\theta}\cos{\phi}\sin{\phi} + \sin^2{\theta}\sin^2{\Phi} \end{array}\right| = \left|\begin{array}{cc}0 & 0 \\\ 0 & 0\end{array}\right|\) This results in a system of equations: \( \begin{cases} \cos^2{\theta}\cos^2{\Phi} + \cos{\theta}\sin{\theta}\cos{\phi}\sin{\phi} = 0 \\ \cos^2{\theta}\cos{\phi}\sin{\phi} + \cos{\theta}\sin{\theta}\sin^2{\Phi} = 0 \\ \cos{\theta}\sin{\theta}\cos^2{\Phi} + \sin^2{\theta}\cos{\phi}\sin{\phi} = 0 \\ \cos{\theta}\sin{\theta}\cos{\phi}\sin{\phi} + \sin^2{\theta}\sin^2{\Phi} = 0 \end{cases} \) We know that \(\cos^2{x} + \sin^2{x} = 1\) for any angle x. So, we can rewrite the first and fourth equations in the system to get expressions for \(\sin{\theta}\cos{\phi}\) and \(\cos{\theta}\sin{\Phi}\) respectively. \( \begin{cases} -\sin{\theta}\cos{\theta}\sin{\phi} = \sin^2{\theta}\cos^2{\Phi} \\ -\sin{\theta}\cos{\theta}\cos{\phi} = \sin^2{\theta}\sin^2{\Phi} \end{cases} \) Now we can divide the second equation by the first equation in the transformed system to find the value of the sine of the difference of angles \(\theta - \Phi\). \(\frac{\sin^2{\theta}\sin^2{\Phi}}{\sin^2{\theta}\cos^2{\Phi}} = \sin^2{\theta - \Phi}\)

Determine the value of \(\theta - \Phi\)

We can cancel out \(\sin^2{\theta}\) from both the numerator and denominator, and we get: \(\frac{\sin^2{\Phi}}{\cos^2{\Phi}} = \sin^2{\theta - \Phi}\) Now, observing the given options, only double-odd multiples of \((\pi / 2)\) produce non-zero values for the given expression. Hence, the correct answer is: \( \theta - \Phi = (2n + 1)(\pi / 2) \Rightarrow \boxed{\text{(b)}} \)

Key concepts

Matrix Multiplication

When we talk about matrix multiplication, it's important to understand that it involves a specific rule set that is quite different from simple scalar multiplication. The fundamental process involves taking the rows of the first matrix and the columns of the second matrix and performing a series of multiplications and additions.

For the given exercise, we considered a 2x2 matrix involving trigonometric functions. To multiply these, we took the cosine and sine values from the first matrix and multiplied them, element-wise, with the corresponding cosine and sine values from the second matrix. This operation requires us to be careful and precise, as each element of the resulting matrix is the sum of products from rows of the first matrix and columns of the second matrix.

• The element in the first row and first column of the product matrix is the sum of the products of the first row of the first matrix and the first column of the second matrix.
• The same principle applies to the other elements of the resultant matrix.

Grasping the pattern of matrix multiplication can significantly simplify the process and reduce errors when dealing with more complex matrices or higher dimensions.

Trigonometric Identities

Understanding trigonometric identities is crucial when dealing with determinants of trigonometric matrices. Trigonometric identities are equations involving trigonometric functions that are true for any value of the involved variables. Probably the most important trigonometric identity is the Pythagorean identity, which states that \(\cos^2{x} + \sin^2{x} = 1\) for any angle x.

This identity simplifies the steps needed to solve trigonometric equations as it allows us to express one trigonometric function solely in terms of another, aiding in the reduction of complex expressions. In the context of our exercise, the Pythagorean identity helps in transforming the system of equations derived from the determinant condition to find the relationship between the angles \(\theta\) and \(\Phi\).

System of Equations

The system of equations emerges when we are dealing with multiple equations that are meant to be solved together, because they share common variables. In the context of matrix multiplication of trigonometric matrices, once we've multiplied the matrices, we're often left with a system of trigonometric equations that must be solved simultaneously.

In this case, the product of our matrices resulted in four equations, each equated to zero. This system of equations can be tricky to solve because they involve trigonometric functions, but using identities and algebraic manipulation simplifies the problem.

To solve the system derived in our exercise, we utilized trigonometric identities which enabled us to condense the system into a single equation that described the relationship between the angles \(\theta\) and \(\Phi\). The approach often involves canceling terms, factoring, and recognizing patterns that align with known identities, which speaks to the interrelatedness of algebra and trigonometry in such problems.