Problem 82 - Precalculus with Limits

Question

In Exercises 75 - 84, find all solutions of the equation in the interval \( \left[0,2\pi\right) \). \( \tan(x + \pi) + 2 \sin(x + \pi) = 0 \)

Step-by-Step Solution

Answer

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

Step 1: Simplify using Trigonometric Identities

Use the identities \(\tan(a + b) = \tan(a) + \tan(b)\) and \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\) and apply it to the given equation. Since \(\tan(\pi) = 0\) and \(\sin(\pi) = 0\), we simplify the equation to: \[\tan(x) + 2 \sin(x) = 0\]

Step 2: Factor the Equation

Rewrite \(\tan(x)\) as \(\frac{\sin(x)}{\cos(x)}\) and rearrange the equation: \[\frac{\sin(x)}{\cos(x)} + 2\sin(x) = 0 \implies \sin(x) (\frac{1}{\cos(x)} + 2) = 0\]

Step 3: Solve for x

Now, set each part in parentheses to zero and solve for \(x\): \[\sin(x) = 0 \implies x = 0, \pi \] \[\frac{1}{\cos(x)} + 2 = 0 \implies \cos(x) = -\frac{1}{2} \implies x = \frac{2\pi}{3}, \frac{4\pi}{3}\]

Step 4: Combine Solutions

The solutions for \(x\) are: \(0, \pi, \frac{2\pi}{3}, \frac{4\pi}{3}\). Since these all fall within our given interval \([0,2\pi)\), these are our solutions.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical expressions that help simplify trigonometric equations. They are formulas involving angles and their trigonometric functions that hold true for all angle measures. For our problem, we use the identities for tangent and sine functions to simplify the given equation. When dealing with the expression \(\tan(x + \pi) + 2 \sin(x + \pi) = 0\), it's crucial to recognize the periodic properties of the tangent and sine functions. We know that \(\tan(\pi) = 0\) and \(\sin(\pi) = 0\). By applying these identities, the equation reduces to \(\tan(x) + 2\sin(x) = 0\).
This reduced form is much simpler and makes solving the equation more manageable. Understanding these identities will also allow you to simplify and solve a wide variety of similar trigonometric equations.

Solution Techniques

To solve trigonometric equations effectively, having a strategic approach is key. Here, the equation \(\tan(x) + 2\sin(x) = 0\) must be manipulated strategically.

One technique is to express trigonometric functions in terms of others. Notice the relation \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This is a useful starting point. By rewriting \(\tan(x)\) and substituting into the equation, we recognize a common factor can be factored out, allowing us to set each factor to zero separately.

• Evaluate both the expressions \(\sin(x) = 0\) and \(\frac{1}{\cos(x)} + 2 = 0\) independently.
• Solve these simpler equations to find possible values of \(x\).

These systematic techniques help narrow down the solutions efficiently, offering a clear path towards solving more complex equations.

Interval Notation

Interval notation is a concise way to express ranges of values, commonly used in specifying solutions to equations. In trigonometric problems, we often deal with cyclical values where intervals specify valid angle measures.
The interval \([0, 2\pi)\) indicates that solutions must be within these bounds, including 0 but excluding \(2\pi\).

When solving \(\tan(x) + 2\sin(x) = 0\), after identifying possible solutions, we must ensure each falls within this specified interval. Thus, our valid solutions for \(x\) are \(0\), \(\pi\), \(\frac{2\pi}{3}\), and \(\frac{4\pi}{3}\), all of which satisfy \([0, 2\pi)\). Understanding interval notation helps you correctly interpret and express the final set of solutions.

Factoring Equations

Factoring is a fundamental technique in solving equations, including those involving trigonometric functions. The process involves expressing an equation as a product of its factors, which simplifies the problem by breaking it down into simpler parts.
In the equation \(\frac{\sin(x)}{\cos(x)} + 2\sin(x) = 0\), we group terms to factor out the common factor \(\sin(x)\). This transforms our equation into:
\[\sin(x)\left(\frac{1}{\cos(x)} + 2\right) = 0\]
The equation now reveals two solvable parts:\

• \(\sin(x) = 0\), which is straightforward to solve;
• \(\frac{1}{\cos(x)} + 2 = 0\), which is simplified to \(\cos(x) = -\frac{1}{2}\).

Factoring helps identify the zero products that yield valid solutions. This technique is powerful as it reduces complex trigonometric relationships into basic identities, allowing for easier resolution.