If 0<θ<πand αdenotes the value of θfor which the system of equations (sinθ) x+y+z=0 x

step 1 So in this problem three linear equations are given and it is also given that theta belongs to 0

If 0<θ<πand αdenotes the value of θfor which the system of...

If $0<\theta<\pi$ and $\alpha$ denotes the value of $\theta$ for which the system of equations
$$
\begin{array}{r}
(\sin \theta) x+y+z=0 \\
x+(\cos \theta) y+z=0 \\
(\sin \theta) x+(\cos \theta) y+z=0
\end{array}
$$
has a non trivial solution, then $2 \sin \alpha+3 \cos \alpha$ is

Key Concepts

Homogeneous Linear Systems

A homogeneous linear system is a set of linear equations in which all the constant terms are zero. Such systems always admit the trivial solution (all variables equal to zero), but they admit nontrivial solutions when the equations are linearly dependent, meaning that the determinant of the system's coefficient matrix is zero.

Determinant Condition for Nontrivial Solutions

In a system of linear equations represented in matrix form, the existence of nontrivial solutions for a homogeneous system is equivalent to the determinant of the coefficient matrix being zero. This condition mathematically assures that the matrix does not have full rank, allowing infinitely many solutions along a subspace.

Trigonometric Functions in Equations

Trigonometric functions like sine and cosine often appear in algebraic equations, introducing oscillatory behavior and periodicity. Their inherent identities and relationships, such as the Pythagorean identity, can be used to simplify the analysis or solution of equations that mix linear algebra with trigonometric expressions.

Solving Trigonometric Equations

The process of solving trigonometric equations involves finding angles within a given interval that satisfy the equation. This typically requires employing trigonometric identities, understanding the periodic nature of the functions involved, and in some cases, applying inverse trigonometric functions to isolate the variable.

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