Problem 64 - Algebra and Trigonometry

Question

Use an addition or subtraction formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi) .\) $$\sin 3 x \cos x-\cos 3 x \sin x=0$$

Step-by-Step Solution

Answer

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

Step 1: Identify Trigonometric Identity

Notice that the expression on the left-hand side of the equation \(\sin 3x \cos x - \cos 3x \sin x = 0\) is a form of the sine addition formula: \(\sin A \cos B - \cos A \sin B = \sin(A - B)\). In our case, \(A = 3x\) and \(B = x\), so we can rewrite the equation as \(\sin(3x - x) = \sin(2x)\).

Step 2: Simplify the Equation

Use the identified identity to simplify the equation: \(\sin(2x) = 0\).

Step 3: Solve for \(x\)

To solve \(\sin(2x) = 0\), find the values of \(2x\) that make \(\sin(2x) = 0\). The sine function equals zero at integer multiples of \(\pi\), i.e., \(2x = n\pi\), where \(n\) is an integer.

Step 4: Solve for \(x\) within the Interval

Since we are looking for solutions in the interval \([0, 2\pi)\), solve \(2x = n\pi\) for \(x\): - \(x = \frac{n\pi}{2}\).Ensure \(x\) is within the specified interval: - For \(n = 0\), \(x = 0\).- For \(n = 1\), \(x = \frac{\pi}{2}\).- For \(n = 2\), \(x = \pi\).- For \(n = 3\), \(x = \frac{3\pi}{2}\).- Skip \(n = 4\) as \(x = 2\pi\), which is not in the interval \([0, 2\pi)\).

Step 5: Confirm All Solutions

The solutions for the equation \(\sin 3x \cos x - \cos 3x \sin x = 0\) in the interval \([0, 2\pi)\) are \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).

Key Concepts

Addition and Subtraction Formulas

In trigonometry, addition and subtraction formulas are indispensable tools. These formulas help in simplifying complex expressions and solving equations. One of the key formulas we use is the sine addition formula, which is expressed as:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

The exercise utilizes the sine subtraction formula. By identifying the expression \( \sin 3x \cos x - \cos 3x \sin x = 0 \) as \( \sin(3x - x) \), we simplify it to \( \sin(2x) \). This transformation makes the equation easier to solve. Breaking down expressions using these identities can significantly simplify complex trigonometric equations.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental trigonometric function. It represents the vertical coordinate of a point on the unit circle. A key property of the sine function is its periodicity:

• It repeats every \( 2\pi \) radians.
• Its values range between -1 and 1.

In the exercise, once we simplify the equation to \( \sin(2x) = 0 \), we rely on these properties to solve for \( x \). Recognizing where the sine function equals zero (such as \( 0, \pi, 2\pi, \) etc.) helps in finding solutions to equations involving sine.

Interval [0, 2π)

When solving trigonometric equations, it is important to focus on the given interval. In this context,

• \([0, 2\pi)\) means we consider values of \( x \) starting from 0 up to, but not including, \( 2\pi \).
• The notation \( [ \) and \( ) \) indicates inclusive and exclusive bounds, respectively.

This restriction helps in narrowing down possible solutions to those that are meaningful and practical for the problem. By using the interval [0, 2\pi), we find solutions that complete one full cycle around the unit circle, ensuring that we account for all possible angles within this range.

Solving Trigonometric Equations

Solving trigonometric equations often involves a few key steps: simplifying the equation, finding where the function equals a particular value, and ensuring the solutions fit within a desired interval. For the provided exercise, these steps can be outlined as:

• Use identities to simplify the equation, such as applying the sine subtraction formula.
• Determine the general solutions, like finding when \( \sin(2x) = 0 \).
• Adjust solutions to fit within the specified interval, ensuring no values are repeated or go beyond the interval \([0, 2\pi)\).

This process involves understanding function properties, utilizing formulas, and careful consideration of the given ranges. Each trigonometric equation might require different strategies, but the core approach remains the same.