Problem 26 - Precalculus with Limits

Question

In Exercises 19-28, find the exact solutions of the equation in the interval \( [0, 2\pi) \). \( (\sin 2x + \cos 2x)^2 = 1 \)

Step-by-Step Solution

Answer

The solutions of the equation \( (\sin 2x + \cos 2x)^2 = 1 \) in the interval \( [0, 2\pi) \) are \( x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\).

Step 1: Simplify the equation

We begin by simplifying the given equation. It's a square on the left side, so we can apply the square of a binomial formula: \( (a + b)^2 = a^2 + 2ab + b^2 \). Therefore, the equation becomes \( (\sin^2 2x + 2 \sin 2x \cos 2x + \cos^2 2x) = 1 \).

Step 2: Apply trigonometric identities

Notice that the left side of the equation now consists of \( \sin^2 a \) and \( \cos^2 a \) where \( a = 2x \), which according to Pythagorean identity sums up to 1, and we have 2 times the product of sine and cosine of the same angle, which is equal to sine of twice that angle. Therefore, our equation becomes \( 1 + \sin 4x) = 1 \).

Step 3: Simplify and solve

Subtract 1 from both sides: \( \sin 4x = 0 \). Solve for \( x \): the solutions in the interval \([0, 2\pi)\) are those for which 4x is an integer multiple of \(\pi\). Therefore, \( x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools for simplifying and solving trigonometric equations. They are equations involving trigonometric functions that hold true for all angles. In this exercise, we use two primary identities:

• Pythagorean Identity: This states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). It helps simplify expressions involving squares of sine and cosine.
• Double Angle Identity for Sine: This is expressed as \( \sin 2\theta = 2\sin \theta \cos \theta \). This identity was used in the step-by-step solution to transform the middle term of the expanded equation into a simpler form.

The proper application of these identities allows us to convert complex trigonometric expressions into manageable forms that can be easily solved, as demonstrated in the solution process.

Solving Equations

Solving trigonometric equations involves finding values of the variable that make the equation true. When solving, it is crucial to work within the given interval, \( [0, 2\pi) \) in this case.

Initially, simplifying the equation using identities is key. As seen in the original solution, the equation was simplified to \( \sin 4x = 0 \), making it easier to solve. The solutions of \( \sin \theta = 0 \) are the angles where the sine function equals zero, which are multiples of \( \pi \):

• \( 0 \)
• \( \pi \)
• \( 2\pi \)

By equating the multiples of \( 4x \) to these zero points and then dividing by 4, the solutions for \( x \) in the specified interval were found. Thoroughly understanding how to isolate \( x \) and restrict solutions to the interval given is crucial in trigonometry.

Unit Circle

The unit circle is a fundamental concept in trigonometry, offering a visual framework to understand angles and corresponding trigonometric function values. A circle with radius 1 centered at the origin of a coordinate plane, it provides a comprehensive picture of angles measured in radians.

Key angles on the unit circle, like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), correspond to the maximum and minimum values of sine and cosine functions. For \( \sin 4x = 0 \), we're interested in angles that lead \( \sin \theta \) to be zero. These angles correspond to the "zero points" on the unit circle:

• The x-axis (\( 0 \) and \( \pi \))
• Multiple full rotations, such as \( 2\pi \)

The unit circle helps in translating what 4x should be to result in those angles and subsequently solving for \( x \) within the interval \( [0, 2\pi) \). Understanding the unit circle facilitates the visualization and solution of trigonometric equations.

Pythagorean Identity

The Pythagorean identity is a crucial element in simplifying trigonometric expressions and solving equations. It conveys the inherent relationship between the sine and cosine of a given angle \( \theta \):

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

In the given equation, \( (\sin 2x + \cos 2x)^2 = 1 \), recognizing that \( \sin^2 2x + \cos^2 2x = 1 \) allowed us to simplify the expression before solving it. This identity shows how the square of sine and cosine sums to a constant value. Thanks to its simplicity, it is a mainstay in solving trigonometric equations, as it helps break down complex expressions into simpler components that can be effectively handled. In practice, recognizing and utilizing such identities allows us to streamline the solving process efficiently. By using the Pythagorean identity, we managed to reduce the original complex equation to a manageable form involving \( \sin 4x \).