Problem 42 - Algebra and Trigonometry

Question

\(39-42\) . Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi) .\) \(\sin 3 \theta \cos \theta-\cos 3 \theta \sin \theta=0\)

Step-by-Step Solution

Answer

The solutions are \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).

Step 1: Identify the Formula

The given expression is \(\sin 3\theta \cos \theta - \cos 3\theta \sin \theta = 0\). This matches the form of the sine of a difference identity, \(\sin(A-B) = \sin A \cos B - \cos A \sin B\). Thus, we can set \(A = 3\theta\) and \(B = \theta\).

Step 2: Apply the Sine Difference Formula

According to the identity \(\sin(A-B) = \sin A \cos B - \cos A \sin B\), the expression \(\sin 3\theta \cos \theta - \cos 3\theta \sin \theta\) simplifies to \(\sin(3\theta - \theta) = \sin(2\theta)\).

Step 3: Set Up the Equation

Now that the expression is simplified to \(\sin(2\theta)\), we set the equation \(\sin(2\theta) = 0\), because the original expression was equal to zero.

Step 4: Find Solutions for Sine Equation

To solve \(\sin(2\theta) = 0\) in the interval \([0, 2\pi)\), we need to find where \(2\theta = n\pi\), where \(n\) is an integer. Solving for \(\theta\) gives us \(\theta = \frac{n\pi}{2}\).

Step 5: Select Solutions in Given Interval

Substitute different values of \(n\) (0, 1, 2, 3, etc.) to find \(\theta\) in \([0, 2\pi)\). This gives the valid solutions \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\). Each is within the specified interval.

Key Concepts

Sine Difference Formula

The Sine Difference Formula is a helpful tool in trigonometry for simplifying and solving trigonometric equations. This formula is expressed as \( \sin(A-B) = \sin A \cos B - \cos A \sin B \). It allows us to rewrite an expression involving a difference of angles in terms of the sine function.
In the given exercise, the trigonometric expression \( \sin 3\theta \cos \theta - \cos 3\theta \sin \theta = 0 \) is recognized to match this formula. By identifying \( A = 3\theta \) and \( B = \theta \), the equation can be simplified to \( \sin(3\theta - \theta) = \sin(2\theta) \).
This transformation reduces the complexity of the expression and makes further work more manageable. Always look for opportunities to use trigonometric identities, like the Sine Difference Formula, to simplify expressions.

Trigonometric Equations

Trigonometric equations are mathematical equations involving trigonometric functions such as sine, cosine, and tangent. Solving these equations often requires using trigonometric identities or constructing simpler equations through mathematical transformations.
In this exercise, once we applied the Sine Difference Formula to simplify to \( \sin(2\theta) = 0 \), our task shifted to solving for the angle \( \theta \).

• First, note that \( \sin x = 0 \) when \( x = n\pi \), where \( n \) is an integer.
• Translate this to our equation, \( 2\theta = n\pi \), which solves to \( \theta = \frac{n\pi}{2} \).

Understanding the generation and solution process for trigonometric equations is critical in solving more complex problems. By breaking down the main equation into simpler parts, solving becomes more straightforward.

Solutions in Interval

The final goal in many trigonometric problems is finding solutions within a specific interval. In this problem, we needed all solutions for \( \theta \) in the interval \([0, 2\pi)\).
Steps for finding solutions include:

• After solving \( \theta = \frac{n\pi}{2} \), we test values of \( n \) (like 0, 1, 2, 3, etc.).
• Ensure the resulting \( \theta \) values fit within the specified range.

In our specific problem, following these steps provided the solutions \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
When tasked with finding solutions, ensure you understand how to adjust \( n \) so that all resulting solutions lie within the required interval. This involves substituting possible whole numbers for \( n \) and checking their corresponding solutions.