Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{9}, \frac{2\pi}{9}, \frac{7\pi}{9}, \frac{8\pi}{9}, \frac{13\pi}{9}, \frac{14\pi}{9} \).

Step 1: Use Angle Sum Identity for Sine

The equation is \( \sin 2x \cos x + \cos 2x \sin x = \sqrt{3}/2 \). Recognize the left side as an angle sum formula for sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Here, applying this identity gives \( \sin((2x) + x) = \sin 3x \). This simplifies the equation to \( \sin 3x = \sqrt{3}/2 \)

Step 2: Solve for 3x

We have the equation \( \sin 3x = \sqrt{3}/2 \). The solutions where \( \sin \theta = \sqrt{3}/2 \) are \( \theta = \frac{\pi}{3} + 2k\pi \) and \( \theta = \frac{2\pi}{3} + 2k\pi \), where \( k \) is an integer. Thus, \( 3x = \frac{\pi}{3} + 2k\pi \) and \( 3x = \frac{2\pi}{3} + 2k\pi \).

Step 3: Solve for x

Now, solve for \( x \) by dividing each part of the equations by 3. Thus, \( x = \frac{\pi}{9} + \frac{2k\pi}{3} \) and \( x = \frac{2\pi}{9} + \frac{2k\pi}{3} \). We need to find values of \( x \) in the interval \([0, 2\pi)\).

Step 4: Find Solutions in [0, 2π)

Substitute integer values of \( k \) to find solutions for \( x \):1. \( k = 0 \): - \( x = \frac{\pi}{9} \) and \( x = \frac{2\pi}{9} \).2. \( k = 1 \): - \( x = \frac{\pi}{9} + \frac{2\pi}{3} = \frac{7\pi}{9} \) and \( x = \frac{2\pi}{9} + \frac{2\pi}{3} = \frac{8\pi}{9} \).3. \( k = 2 \): - \( x = \frac{\pi}{9} + \frac{4\pi}{3} = \frac{13\pi}{9} \) and \( x = \frac{2\pi}{9} + \frac{4\pi}{3} = \frac{14\pi}{9} \).4. \( k = 3 \): - \( x = \frac{\pi}{9} + \frac{6\pi}{3} = \frac{19\pi}{9} \) which is more than \( 2\pi \), so it is not valid. The solutions within \([0, 2\pi)\) are \( x = \frac{\pi}{9}, \frac{2\pi}{9}, \frac{7\pi}{9}, \frac{8\pi}{9}, \frac{13\pi}{9}, \frac{14\pi}{9} \).

Key Concepts

Angle Sum Identity

The angle sum identity is a crucial trigonometric identity used to simplify complex expressions involving angles. It states that for any angles \( a \) and \( b \), the sine of their sum can be expressed using:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

In our problem, the expression \( \sin 2x \cos x + \cos 2x \sin x \) matches the form \( \sin a \cos b + \cos a \sin b \). By recognizing this, it can be rewritten using the angle sum identity as \( \sin(2x + x) = \sin 3x \). This simplifies the problem significantly by reducing it to a single simpler equation: \( \sin 3x = \sqrt{3}/2 \). Understanding this identity is essential for algebraically managing and manipulating trigonometric equations.

Solving Trigonometric Equations

When solving trigonometric equations such as \( \sin 3x = \sqrt{3}/2 \), it's crucial to find all potential solutions. The process typically involves:

• Identifying Known Angles: Where the trigonometric function equals the known value, like \( \theta = \frac{\pi}{3} \).
• Applying Formula: The general solution for \( \sin \theta = \sqrt{3}/2 \) includes angles that take the form \( \theta = \frac{\pi}{3} + 2k\pi \) or \( \theta = \frac{2\pi}{3} + 2k\pi \) for integers \( k \).
• Adjusting the equation: Replace \( \theta \) with \( 3x \) since the angle is expressed in terms of \( x \).

This results in the equations \( 3x = \frac{\pi}{3} + 2k\pi \) and \( 3x = \frac{2\pi}{3} + 2k\pi \). Step by step, these equations help us find solutions for \( x \) that satisfy the initial conditions.

Interval Solutions

Finding solutions within a specific interval involves narrowing down the general solutions of a trigonometric equation to those that fit within the given limits. In this case, we want solutions for \( x \) within \( [0, 2\pi) \). Here's how it works:

• Determine the range for \( x \) by dividing the solutions for \( 3x \) by 3. This results in \( x = \frac{\pi}{9} + \frac{2k\pi}{3} \) and \( x = \frac{2\pi}{9} + \frac{2k\pi}{3} \).
• Select integer values for \( k \) (e.g., \( k = 0, 1, 2, \ldots \)) and solve for \( x \).
• Verify each calculated \( x \) to ensure it lies within \( [0, 2\pi) \).

By doing this, you filter out non-valid solutions, ensuring all solutions comply with the problem's given interval.

Sine Function

The sine function is one of the fundamental trigonometric functions, commonly used for expressing periodic phenomena. Here's what to keep in mind:

• The range of the sine function is from -1 to 1. Hence, it can never exceed these values, defining limits for possible solutions.
• It repeats every \( 2\pi \), known as the period. This periodic nature is essential when determining full sets of solutions within an interval.
• Common angles for sine are often memorized, for instance \( \sin(\frac{\pi}{6}) = \frac{1}{2} \) and \( \sin(\frac{\pi}{3}) = \sqrt{3}/2 \), which are highly useful for solving equations quickly.

In this example, knowing that \( \sin 3x = \sqrt{3}/2 \) lets us quickly identify the vital angles that satisfy the equation, which directly aids in solving our problem.