Problem 45 - Precalculus: Functions and Graphs

Question

Solve the system for \(a\) and \(b\). (Hint: Treat terms such as \(e^{3 x}, \cos x,\) and \(\sin x\) as "constant coefficients.") $$\left\\{\begin{aligned} a \cos x+b \sin x &=0 \\ -a \sin x+b \cos x &=\tan x \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \( a = -\tan x \) and \( b = 1 \).

Step 1: Identify the System of Equations

We are given two equations: \[ a \cos x + b \sin x = 0 \] and \[ -a \sin x + b \cos x = \tan x \].Our goal is to solve for \( a \) and \( b \).

Step 2: Express \( b \) in terms of \( a \)

From the first equation \( a \cos x + b \sin x = 0 \), solve for \( b \):\[ b \sin x = -a \cos x \]\[ b = -\frac{a \cos x}{\sin x} \]\[ b = -a \cot x \].

Step 3: Substitute \( b \) in the Second Equation

Substitute \( b = -a \cot x \) into the second equation, \( -a \sin x + b \cos x = \tan x \): \[ -a \sin x + (-a \cot x) \cos x = \tan x \]Simplify the expression: \[ -a \sin x - a \cos x \cot x = \tan x \].

Step 4: Simplify and Solve for \( a \)

Rewrite \(-a \cos x \cot x\) using the identity \( \cot x = \frac{\cos x}{\sin x} \):\[ -a \sin x - a \frac{\cos^2 x}{\sin x} = \tan x \]Combine terms:\[ a(-\sin^2 x - \cos^2 x) = \tan x \]Note that \(-\sin^2 x - \cos^2 x = -1\):\[ -a = \tan x \]Thus, \[ a = -\tan x \].

Step 5: Substitute Back to Find \( b \)

Substitute \( a = -\tan x \) into the expression for \( b \):\[ b = -(-\tan x) \cot x \]Since \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \), then:\[ b = \frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} \]\[ b = 1 \].

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that help in simplifying and solving equations that involve trigonometric functions like sine, cosine, and tangent. These identities provide relationships between different trigonometric ratios. Some of the most common ones include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \).
• Reciprocal Identities: \( \csc x = \frac{1}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), \( \cot x = \frac{1}{\tan x} \).
• Tangent and Cotangent Identities: \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

In the problem at hand, we see the application of the tangent and cotangent identities, especially where \( \tan x \) is expressed as \( \frac{\sin x}{\cos x} \) and \( \cot x \) as \( \frac{\cos x}{\sin x} \). This conversion allows us to simplify expressions effectively, especially when the identities can be substituted back into equations, revealing relationships that are less obvious at first glance. Recognizing and applying these identities accurately is a step towards successfully solving systems that include trigonometric terms.

Substitution Method

The substitution method is a fundamental technique used in algebra to solve systems of equations. The goal is to express one variable in terms of another, which then allows it to be substituted into a second equation. This reduces the system to a single equation with one variable, which is easier to solve.
Here's how the substitution method works:

• Solve one equation for one variable: In our problem, we start by solving the equation \( a \cos x + b \sin x = 0 \) for \( b \), which gives us \( b = -a \cot x \).
• Substitute into the second equation: We then substitute the expression for \( b \) into the second equation, creating a new equation in terms of \( a \).
• Solve the reduced equation: The resulting equation is simplified, allowing us to solve for \( a \).

This approach allows us to focus on one variable at a time, making the system less complex. It systematically narrows down the possibilities, checking off each variable until a solution is achieved. By maintaining careful calculation, the substitution method simplifies systems that may seem intricate at first glance.

Solving Equations

The core goal in solving equations is to find the values of the variables that satisfy all given equations concurrently. In our particular exercise, it's about finding the values for \( a \) and \( b \) in the context of trigonometric expressions.
When solving equations involving trigonometric terms:

• Recognize constants and variable coefficients: Here, terms like \( \cos x \), \( \sin x \), and \( \tan x \) are treated as constants with respect to \( a \) and \( b \).
• Simplify using identities: The problem involves mindful application of trigonometric identities to simplify expressions. For example, using \( \tan x = \frac{\sin x}{\cos x} \).
• Logical progression: Move through equations in a logical fashion, solving step by step rather than attempting to tackle everything at once.

Eventually, after substitution and simplification, you land on values for \( a \) and \( b \). Here, applying identities is critical as it cuts down complexity, leading to the conclusion that \( a = -\tan x \) and \( b = 1 \). Careful analysis and procedural methods often clear the path to finding solutions in systems of equations.