If $$ \mathrm{D}=\left|\begin{array}{ccc} 1 & 3 \cos \theta & 1 \\ \sin \theta & 1 & 3 \cos \theta \\ 1 & \sin \theta & 1 \end{array}\right| $$ then maximum value of \(\mathrm{D}\) is... (a) 9 (b) 1 (c) 10 (d) 16

Determinants play a critical role in linear algebra and are used to compute the solutions of a system of linear equations, evaluate eigenvalues, and assist in understanding the geometry of transformations. The determinant of a matrix provides valuable information about the system it represents; for instance, a non-zero determinant suggests the matrix is invertible, while a zero determinant indicates the system has either no solutions or infinitely many solutions.

When analyzing a determinant, it is often useful to expand it across a row or column. The determinant of a 3x3 matrix, as seen in the exercise, can be expanded in terms of 2x2 sub-determinants - this method is known as the Laplace Expansion. Through simplifying sub-determinants and combining like terms, the original determinant's value can be found efficiently.

Finding the maximum value of a determinant that involves trigonometric functions typically involves leveraging the properties of sines and cosines to simplify the expression. If a determinant's maximum value depends on an angle, like \(\theta\), it's beneficial to examine the behavior of trigonometric functions to find the extreme values.

In our case, after the determinant is expanded and simplified, we optimize the trigonometric expressions to obtain the maximum value. Since \(\sin^2\theta\) peaks at 1 and both \(\sin\theta\) and \(\cos\theta\) range between -1 and 1, these bounds are used to pinpoint the condition where other terms cancel out or reinforce one another to reach an extremum. The exercise demonstrates a situation where the maximum determinant value is achieved when certain trigonometric functions take specific values that nullify the impact of other terms.

The sine and cosine functions are fundamental in trigonometry, and they describe the ratios of the sides of a right triangle relative to one of the non-right angles. They have a range of values from -1 to 1 and period of \(2\pi\), indicating that they repeat their values every \(2\pi\) radians. Moreover, they are continuous and differentiable, allowing us to find their maximum and minimum values easily through calculus if necessary.

The properties of these functions are especially useful when solving trigonometric expressions within determinants. For example, we know that \(\sin^2\theta + \cos^2\theta = 1\), which helps simplify and solve for angles that yield the maximum or minimum values of expressions. In the case of the exercise, understanding these properties is crucial to determine when the product of sine and cosine terms will lead to their respective maximum or null values, aiding in the calculation of the determinant's maximum value.