Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \tan \alpha+\tan ^{2} \alpha=0 $$

Step-by-Step Solution

Answer

The solutions are \( \alpha = n\pi \) and \( \alpha = -\frac{\pi}{4} + n\pi \), where \( n \) is an integer.

Step 1: Set the Equation to Zero

Start by setting the equation equal to zero: \( \tan \alpha + \tan^2 \alpha = 0 \).

Step 2: Factor the Expression

Factor out \( \tan \alpha \) from the equation: \( \tan \alpha (1 + \tan \alpha) = 0 \).

Step 3: Solve Each Factor

Set each factor equal to zero to find the solutions. For \( \tan \alpha = 0 \), the solutions are \( \alpha = n\pi \), where \( n \) is an integer. For \( 1 + \tan \alpha = 0 \), solve \( \tan \alpha = -1 \), yielding \( \alpha = -\frac{\pi}{4} + n\pi \), where \( n \) is an integer.

Step 4: Combine Solutions

Combine the solutions from both factors. The general solutions are \( \alpha = n\pi \) and \( \alpha = -\frac{\pi}{4} + n\pi \), where \( n \) is any integer.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions, alongside sine and cosine. It is represented by \( \tan \alpha \). The tangent of an angle in a right triangle can be defined as the ratio of the opposite side to the adjacent side. In a unit circle, it represents the y-coordinate divided by the x-coordinate for a given angle from the positive x-axis.

Here are some important properties of the tangent function:

• Periodicity: The basic period of the tangent function is \( \pi \), meaning it repeats every \( \pi \) radians.
• Asymptotes: The function has vertical asymptotes where the cosine of the angle is zero.
• Odd function: This means that \( \tan(-\alpha) = -\tan(\alpha) \).

For solving equations involving the tangent function, it's crucial to know these properties. This helps when determining the general solutions over all real numbers.

Factoring Equations

Factoring is a critical algebraic process used to simplify and solve equations. It involves expressing an equation as a product of its factors, revealing solutions that are easier to find.

In the context of trigonometric equations like \( \tan \alpha + \tan^2 \alpha = 0 \), factoring helps break down a complex expression into simpler parts. In this case, we factor it as \( \tan \alpha (1 + \tan \alpha) = 0 \).

• Identify Common Factors: Recognize that \( \tan \alpha \) is common in both terms.
• Break Down Expression: By factoring out \( \tan \alpha \), the equation is simplified to a multiplication of terms.
• Apply Zero Product Property: Set each factor to zero to solve each part: \( \tan \alpha = 0 \) and \( 1 + \tan \alpha = 0 \).

This approach reveals solutions systematically and ensures no potential solutions are overlooked.

General Solutions

General solutions provide a complete set of possible solutions for trigonometric equations. They are essential especially when dealing with functions like tangent, which are periodic.

Once we factor the equation \( \tan \alpha (1 + \tan \alpha) = 0 \), we solve individual cases:

• Case 1: \( \tan \alpha = 0 \)
  • The solution for \( \tan \alpha = 0 \) is \( \alpha = n\pi \), capturing its periodic nature with \( \pi \)-intervals.
• Case 2: \( 1 + \tan \alpha = 0 \)
  • Solving \( \tan \alpha = -1 \), we find \( \alpha = -\frac{\pi}{4} + n\pi \), acknowledging the symmetry and periodicity around \( -\frac{\pi}{4} \).

These general solutions encompass all possible angles that satisfy the original equation, accounting for the periodic properties of the tangent function. Integer \( n \) allows the solutions to span the entire set of real numbers.

Integer Solutions

Integer solutions refer to the solutions that depend on integer values, usually denoted by \( n \) in trigonometric general solutions. By incorporating integer multiples, it allows us to capture the periodic nature of trigonometric functions across their domains.

For the equation \( \tan \alpha + \tan^2 \alpha = 0 \), involving integer solutions means:

• The solutions derived from \( \tan \alpha = 0 \) are represented as \( \alpha = n\pi \), with \( n \) as any integer, showing repeats every \( \pi \) radians.
• For \( \tan \alpha = -1 \), the solution \( \alpha = -\frac{\pi}{4} + n\pi \) still relies on \( n \), reflecting repetition every \( \pi \) radians, offset by \( -\frac{\pi}{4} \).

These integer-based solutions highlight the continuous array of angles that solve the initial equation, emphasizing how the periodicity of tangent is explicitly governed by integers.