Problem 25 - Precalculus with Limits

Question

In Exercises 19-28, find the exact solutions of the equation in the interval \( [0, 2\pi) \). \( \sin 4x = - 2 \sin 2x \)

Step-by-Step Solution

Answer

The solutions of the equation \( \sin 4x = - 2 \sin 2x \) in the interval [0, 2π) are \( x = \frac{\pi}{2}, \frac{3\pi}{2} \)

Step 1: Apply Trigonometric Identity

Use the double angle formula \( \sin 2u = 2 \sin u \cos u \) to transform \( \sin 4x \) into \( 2 \sin 2x \cos 2x \). The equation becomes:\[ 2 \sin 2x \cos 2x = - 2 \sin 2x \]

Step 2: Simplify the Equation

Now, divide both sides of the equation by \( \sin 2x \) to get:\[ 2 \cos 2x = -2 \]The left side is simply \( \cos 2x = -1 \]

Step 3: Solve for x

The cosine function equals -1 at \( \pi \), therefore, \[ 2x = \pi, 3\pi \]Which means, \[ x = \frac{\pi}{2}, \frac{3\pi}{2} \]

Key Concepts

Double Angle Formula

The double angle formulas are a set of trigonometric identities that express trigonometric functions of double angles, such as \(2x\), in terms of trigonometric functions of single angles, \(x\). When dealing with sine, the double angle formula states that \(\sin(2x) = 2\sin(x)\cos(x)\). This is particularly useful for simplifying trigonometric expressions or solving equations where the argument of trigonometric functions is a multiple of another angle, as seen in the given exercise.

The double angle formula can also be applied to the cosine and tangent functions. For the cosine, there are two common expressions:

• \(\cos(2x) = \cos^2(x) - \sin^2(x)\)
• \(\cos(2x) = 2\cos^2(x) - 1\)
• \(\cos(2x) = 1 - 2\sin^2(x)\)

These formulas are invaluable in solving complex trigonometric equations and can reduce the equation to a more familiar form where standard trigonometric values can be applied to find exact solutions.

Exact Solutions in Trigonometry

Finding exact solutions in trigonometry involves solving trigonometric equations for specific angle measures, where the solutions are typically represented in terms of \(\pi\) to indicate their exactness rather than using approximate decimal values. Our exercise requires solving for \(x\) within the interval \([0, 2\pi)\), which means finding all possible angles between 0 and 2\(\pi\) (or 0 and 360 degrees) that satisfy the given trigonometric equation.

To obtain exact solutions, one often uses well-known values of trigonometric functions at special angles, such as \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\) (or in radians: 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\)). It's important to consider that trigonometric functions are periodic, meaning they repeat their values at regular intervals. For the cosine function, its period is \(2\pi\), so when finding solutions, one must be aware of this repetition to list all solutions within the specified interval.

Cosine Function

The cosine function is one of the primary trigonometric functions and is often denoted as \(\cos(x)\). It represents the x-coordinate of the point on the unit circle that is reached by traveling x units around the circle, starting from the positive x-axis. The cosine function is an even function, meaning it has symmetry along the y-axis, and it has a range of -1 to 1. In the context of the given exercise, understanding the periodic nature and the symmetry of the cosine function is crucial since this particular problem requires knowledge of where the cosine function equals -1.

The standard values for the cosine function at important angles are essential for solving trigonometric equations. For instance, \(\cos(\pi) = -1\), \(\cos(0) = 1\), and \(\cos(\frac{\pi}{2}) = 0\). These values assist in identifying when the function takes on specific values, influencing the exact solutions we seek. Being able to link these values with the double angle formula helps solve complex trigonometric equations in a more efficient and precise manner.