Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \cot ^{2} x-3=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{6} + k\pi \) and \( x = \frac{5\pi}{6} + k\pi \), \( k \in \mathbb{Z} \).

Step 1: Set the Equation to Zero

First, let's set the equation to zero by rewriting it as \( \cot^2 x - 3 = 0 \).

Step 2: Isolate the Trigonometric Function

To isolate \( \cot^2 x \), add 3 to both sides of the equation: \( \cot^2 x = 3 \).

Step 3: Solve for \( \cot x \)

Take the square root of both sides to solve for \( \cot x \): \( \cot x = \pm \sqrt{3} \).

Step 4: Determine Possible Angles Using \( \cot x = \sqrt{3} \)

For \( \cot x = \sqrt{3} \), recall that \( \cot x = \frac{1}{\tan x} \), which means \( \tan x = \frac{1}{\sqrt{3}} \). This occurs when \( x = \frac{\pi}{6} + k\pi \), where \( k \) is an integer.

Step 5: Determine Possible Angles Using \( \cot x = -\sqrt{3} \)

For \( \cot x = -\sqrt{3} \), recall that \( \tan x = -\frac{1}{\sqrt{3}} \). This occurs when \( x = \frac{5\pi}{6} + k\pi \), where \( k \) is an integer.

Step 6: Combine All Solutions

Therefore, the solutions to the original equation \( \cot^2 x - 3 = 0 \) are \( x = \frac{\pi}{6} + k\pi \) and \( x = \frac{5\pi}{6} + k\pi \), where \( k \) is any integer.

Key Concepts

Cotangent Function

The cotangent function, often denoted as \( \cot x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In mathematical terms, \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \). This means that cotangent gives us the ratio of the adjacent side to the opposite side in a right triangle where the angle is \( x \).
\( \cot x \) is undefined for angles where the sine of the angle is zero, as this would mean division by zero. This occurs, for example, at multiples of \( \pi \) (0, \( \pi \), 2\( \pi \), etc.). Therefore, the cotangent function is periodic with a period of \( \pi \), and is undefined at integer multiples of \( \pi \).

• Period: \( \pi \)
• Undefined: when \( x = k\pi \), where \( k \) is an integer.

The cotangent function is widely used in solving trigonometric equations where understanding its properties allows us to find specific angles that satisfy given conditions.

Solving Equations

To solve trigonometric equations, it's important to first understand the trigonometric function involved. For this particular equation, \( \cot^2 x - 3 = 0 \), we need to isolate \( \cot x \) to find potential solutions. Solving such equations typically involves a few systematic steps:

• Isolate the trigonometric function: Add or subtract values to rearrange the equation.
• Solve algebraically: Use square roots or other algebraic methods like factoring.
• Consider plus/minus solutions when squaring/square rooting: Remember to account for positive and negative solutions.
• Identify matching angles using the unit circle or known values.

In our case, once we have \( \cot^2 x = 3 \), taking the square root gives \( \cot x = \pm \sqrt{3} \). This reduces our task to finding angles where the cotangent is \( \sqrt{3} \) or \(-\sqrt{3} \). Understanding these steps lays the groundwork for solving more complex trigonometric equations you might encounter.

Angle Solutions

When given \( \cot x = \sqrt{3} \), we need to determine which angles satisfy this equation. By referring to common angles and their tangent values, we find that \( \tan x = \frac{1}{\sqrt{3}} \) at \( x = \frac{\pi}{6} \). Therefore, \( \cot x = \sqrt{3} \) at these specific angles. But remember, since cotangent has a period of \( \pi \), we must add \( k\pi \) (where \( k \) is an integer) to cover all possible solutions.
Similarly, \( \cot x = -\sqrt{3} \) occurs when \( x = \frac{5\pi}{6} \), with the same period adjustment.
This leads to:

• \( x = \frac{\pi}{6} + k\pi \), where \( k \) is any integer.
• \( x = \frac{5\pi}{6} + k\pi \), where \( k \) is any integer.

Thus, we can list the entire set of solutions comprehensively by considering both the primary angle and its periodic repetition. This method ensures we capture all possible angles that satisfy the original equation, demonstrating a fundamental technique used in solving trigonometric equations.