91Ó°ÊÓ

Trigonometric properties play a crucial role when solving systems that include angles and lengths, like the system of linear equations involving \(\cos\theta\) and \(\sin\theta\) provided in the exercise. To solve such equations, we must recall key trigonometric identities, such as \(\sin^2\theta + \cos^2\theta = 1\). These properties allow us to simplify complex expressions and, importantly, to manipulate equations to isolate variables.

Other useful tools from trigonometry include the sum and difference formulas, which provide ways to combine or break apart expressions involving \(\sin\theta\) and \(\cos\theta\). These tools are especially useful when we need to add or subtract angles or even to simplify the given equations in the system. Understanding how to apply these properties gives a powerful way to navigate through problems involving trigonometry in linear systems.

A system of linear equations consists of two or more linear equations that have common variables. Our goal when solving such systems is to find the values of these variables that satisfy all the equations simultaneously. Systems can be categorized by the number of solutions they possess: one unique solution, infinitely many solutions, or no solution at all.

In the exercise, we're provided with two equations with two unknowns \(x\) and \(y\), each modified by trigonometric functions of an angle \(\theta\). Solving this system allows us to understand not just the relationship between \(x\) and \(y\), but also their relationship to the angle \(\theta\). The intersection of these equations, if it exists, is the solution to the system.

The substitution method is one of several techniques to solve a system of linear equations. The core idea is to solve one of the equations for one variable and then 'substitute' this expression in the other equation(s). This method is particularly useful when one of the equations can easily be solved for one of the variables.

In the given exercise, after determining the value of \(x\), we substitute it back into one of the original equations to find \(y\). By doing so, we simplify the system to a single variable equation that can be resolved. The substitution method, while straightforward, requires careful algebraic manipulation to ensure accuracy throughout the process.

The elimination method is another strategy to solve systems of linear equations. This approach focuses on adding or subtracting equations from each other to eliminate one of the variables. Strategic multiplication of the entire equation by a number may be necessary before addition or subtraction to align the coefficients and effectively eliminate a variable.

As seen in the exercise, by multipling equations and then subtracting them, we eliminated the \(y\) variable, simplifying the system to a single variable, \(x\). The elimination method is often the preferred choice when the system is set up in a way that allows for a straightforward elimination of variables. It's a powerful tool that, with practice, can be used quickly and efficiently.