Problem 50 - Algebra and Trigonometry

Question

\(39-56 \approx\) Solve the given equation. $$ 3 \tan ^{3} \theta=\tan \theta $$

Step-by-Step Solution

Answer

\(\theta = n\pi, \theta = \frac{\pi}{6} + n\pi, \theta = \frac{5\pi}{6} + n\pi\).

Step 1: Set Equation to Zero

Start with the given equation: \(3 \tan^3 \theta = \tan \theta\). First, bring all terms to one side to set the equation to zero: \(3 \tan^3 \theta - \tan \theta = 0\).

Step 2: Factor the Common Term

Notice that \( \tan \theta\) is a common factor. Factor it out: \(\tan \theta (3 \tan^2 \theta - 1) = 0\).

Step 3: Solve Each Factor Separately

This factorization gives us two equations to solve:1. \(\tan \theta = 0\)2. \(3 \tan^2 \theta - 1 = 0\)

Step 4: Solve \(\tan \theta = 0\)

The first equation, \(\tan \theta = 0\), gives solutions where \(\theta\) is a multiple of \(\pi\), i.e., \(\theta = n\pi\) where \(n\) is an integer.

Step 5: Solve \(3 \tan^2 \theta - 1 = 0\)

Simplify the second equation: \(3 \tan^2 \theta = 1\). Divide both sides by 3: \(\tan^2 \theta = \frac{1}{3}\). Take the square root of both sides: \(\tan \theta = \pm \frac{1}{\sqrt{3}}\).

Step 6: Find Angles for \(\tan \theta = \pm \frac{1}{\sqrt{3}}\)

The angles where \(\tan \theta = \frac{1}{\sqrt{3}}\) are \(\theta = \frac{\pi}{6} + n\pi\). For \(\tan \theta = -\frac{1}{\sqrt{3}}\), the angles are \(\theta = \frac{5\pi}{6} + n\pi\). Here, \(n\) is an integer for both solutions.

Key Concepts

Tangent Function

The tangent function is an essential concept in trigonometry. It is denoted as \(\tan \theta\), and provides the ratio of the length of the opposite side to the adjacent side in a right-angled triangle. The tangent function is unique because it can take on any real number value, making it crucial in solving trigonometric equations.

• In this exercise, the function \(\tan \theta\) is prominently featured, showcasing its versatility.
• For tangent values like \(0\), the corresponding angle \(\theta\) is straightforward to determine, such as \(\theta = n\pi\) for any integer \(n\).
• The general periodicity of the tangent function is \(\pi\), which means its pattern repeats every \(\pi\) radians.

Understanding the tangent function allows you to explore various angle solutions and handle more complex trigonometric equations effectively.

Factoring Equations

Factoring techniques play a critical role in solving equations, particularly polynomial ones. In this exercise, factoring is used to simplify the original trigonometric equation.
Begin with the expression \(3 \tan^3 \theta - \tan \theta = 0\).

• The first step is to identify the common factor, \(\tan \theta\).
• After factoring out \(\tan \theta\), the equation becomes \(\tan \theta (3 \tan^2 \theta - 1) = 0\).

Through factoring, this challenging equation is divided into simpler parts that are easier to solve. Each factor can then be set to zero individually, leading to distinct solutions. Factoring is like breaking down a complex puzzle into manageable pieces.

Angle Solutions

Finding angle solutions requires solving the simplified equations obtained post-factoring. For this problem, you get two separate equations:

• \(\tan \theta = 0\), which provides a straightforward solution: \(\theta = n\pi\).
• \(3 \tan^2 \theta = 1\), leading to \(\tan \theta = \pm \frac{1}{\sqrt{3}}\).

After finding these values for \(\tan \theta\), work out the associated angles using known trigonometric identities.

• For \(\tan \theta = \frac{1}{\sqrt{3}}\), the angles are \(\theta = \frac{\pi}{6} + n\pi\).
• For \(\tan \theta = -\frac{1}{\sqrt{3}}\), you find \(\theta = \frac{5\pi}{6} + n\pi\).

The use of angle solutions is vital, as it represents the method of dealing directly with solutions in their trigonometric context, considering the periodic nature of trigonometric functions.

Trigonometry

Trigonometry is a broad field of mathematics dealing with the relationships between the angles and sides of triangles. It is extensively employed in solving equations like the one in this exercise.

• Trigonometric functions, including tangent, sine, and cosine, relate angles to side ratios.
• The periodic nature of these functions makes them powerful for solving equations with multiple or infinite solutions over a domain.

It is essential to grasp the fundamentals of trigonometry to tackle problems involving periodicity and symmetry of angles. When an equation involves \(\tan \theta\), a strong understanding of trigonometry provides a route to decipher real-world phenomena and mathematical abstractions alike. Recognizing symmetries can simplify problems significantly, offering clear pathways to solutions. Learning trigonometry equips students with the computational tools and logical framework necessary for advanced mathematical reasoning.