Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-48: Solve the system for \(a\) and \(b\). (Hint: Treat terms such as \(e^{3 x}, \cos x\), and \(\sin x\) as "constant coefficients.") $$ \left\\{\begin{array}{c} a \cos x+b \sin x=0 \\ -a \sin x+b \cos x=\sin x \end{array}\right. $$

Step-by-Step Solution

Answer

\(a = -\sin^2 x\); \(b = \sin x \cos x\).

Step 1: Analyze the System of Equations

The given system of equations is \( a \cos x + b \sin x = 0 \) and \( -a \sin x + b \cos x = \sin x \). We need to solve for the variables \( a \) and \( b \).

Step 2: Express One Variable in Terms of the Other

From the first equation, express \( a \) in terms of \( b \): \( a \cos x = -b \sin x \). Thus, \( a = - \frac{b \sin x}{\cos x} \).

Step 3: Substitute into the Second Equation

Substitute the expression for \( a \) from Step 2 into the second equation: \(-\left(- \frac{b \sin x}{\cos x}\right) \sin x + b \cos x = \sin x\). Simplify this to \( \frac{b (\sin^2 x)}{\cos x} + b \cos x = \sin x \).

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

Combine terms to solve for \( b \): \( b \left( \frac{\sin^2 x}{\cos x} + \cos x \right) = \sin x \). Simplify and solve: \( b \left( \frac{\sin^2 x + \cos^2 x}{\cos x} \right) = \sin x \). Using \( \sin^2 x + \cos^2 x = 1 \), this becomes \( b \frac{1}{\cos x} = \sin x \). Therefore, \( b = \sin x \cos x \).

Step 5: Solve for \( a \) Using the Expression for \( b \)

Substitute \( b = \sin x \cos x \) back into the expression for \( a \) from Step 2: \( a = - \frac{\sin x \cos x \sin x}{\cos x} = -\sin^2 x \). Therefore, \( a = -\sin^2 x \).

Step 6: Verify the Solution

Substitute \( a = -\sin^2 x \) and \( b = \sin x \cos x \) back into both original equations to ensure they hold true. The computations confirm the solution is valid.

Key Concepts

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. They appear frequently in problems of geometry and physics.
Understanding how to manipulate these functions is crucial in solving trigonometric equations.Here are a few key points about trigonometric equations:

• They often require special identities, such as the Pythagorean identity: \[\sin^2 x + \cos^2 x = 1\]
• Functions like sine and cosine have specific ranges; for example, both these functions have outputs between -1 and 1.
• Solutions can often be periodic, meaning they repeat at specific intervals.

In the exercise provided, functions such as \(\cos x\) and \(\sin x\) are treated as constant coefficients, helping simplify the system of equations into a more solvable form.

Solving Systems

Solving systems of equations involves finding values that satisfy all given equations simultaneously. Systems can consist of linear equations, non-linear ones, or, as in the current exercise, trigonometric equations.
This exercise involves solving a system of two equations to determine unknowns \(a\) and \(b\).Here's a step-by-step approach:

• Analyze each equation: Observe how variables relate to each other.
• Express one variable in terms of the other: For two variables, express one in terms of the other using one of the equations.
• Substitute and simplify: Replace the expressed variable in the second equation, simplifying to solve for the remaining variable.
• Back-substitute: Having found one variable, substitute back to find the other.

Each step reduces complexity, narrowing down possible values for the variables, like \(a = -\sin^2 x\) and \(b = \sin x \cos x\) in this context.

Algebraic Manipulation

Algebraic manipulation refers to moving and rearranging parts of equations to isolate variables, making solving them easier. This is a key skill in solving not only algebraic but also trigonometric systems of equations.
In this exercise, algebraic manipulation simplifies solving the system of equations.There's a series of common techniques used in algebraic manipulation:

• Substitution: This involves replacing one variable with an expression derived from another equation. This is frequently the first substantial step in solving using algebraic manipulation.
• Simplifying: This can include combining like terms or using identities, such as how we used \(\sin^2 x + \cos^2 x = 1\) to simplify expressions to a solvable form.
• Factorization: Though not used here, it's often employed to break down policies or conditions into manageable factors.

Mastering algebraic manipulation allows tackling even complex systems efficiently, ensuring a smooth path to finding accurate solutions.