Problem 58 - Algebra and Trigonometry

Question

Solve each equation on the interval \([0,2 \pi)\) $$ \cot x(\tan x+1)=0 $$

Step-by-Step Solution

Answer

The solutions of the equation \(\cot x (\tan x + 1) = 0\) on the interval [0, 2pi) are \(x = \frac{\pi}{2}\), \(x = \frac{3\pi}{2}\), \(x = \frac{3\pi}{4}\), and \(x = \frac{7\pi}{4}\)

Step 1: Set each factor to Zero

The equation can be written as: \(\cot x = 0\) and \(\tan x + 1 = 0\), by using the zero-product property.

Step 2: Solve for x in the first Equation

The cotangent function equals zero at \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\), thus the solution for \(\cot x = 0\) in the range \([0, 2\pi)\) are \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\).

Step 3: Solve for x in the second Equation

Solving the second equation \(\tan x + 1 = 0\) for x results into: \(\tan x = -1\). From the unit circle, the solutions for this in the given range are \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\).

Key Concepts

Understanding Cotangent

Cotangent, often denoted as \( \cot \theta \), is a fundamental trigonometric function commonly used in solving equations involving right triangles and periodic phenomena. It is the reciprocal of the tangent function. Mathematically, it's expressed as:\[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]This equation reveals that the cotangent is undefined wherever the sine function equals zero, as division by zero is not possible. Specifically, these undefined points occur wherever the angle is a multiple of \( \pi \). Thus, cotangent is uniquely zero where the cosine function equals zero, notably at angles like \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).

• Unique Points: Cotangent equals zero when cosine equals zero, such as at \( \theta = \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).
• Undefined: Cotangent is undefined at multiples of \( \pi \) where sine equals zero.

Understanding these properties is crucial for solving trigonometric equations, as it asks us to identify and leverage intervals where certain trigonometric functions get specific values. Indeed, the cotangent plays a vital role in trigonometry, intricately linked to both sine and cosine.

Exploring Tangent

Tangent is another essential trigonometric function, commonly written as \( \tan \theta \). It represents the ratio of the opposite side to the adjacent side in a right triangle, or equivalently, the sine over the cosine:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This equation indicates that the tangent is undefined at points where the cosine function equals zero, highlighting discontinuities at odd multiples of \( \frac{\pi}{2} \). Unlike cotangent, the tangent is defined to be zero when sine equals zero.

• Zero Points: The tangent function is zero where sine is zero, such as at \( \theta = 0, \pi, 2\pi \), etc.
• Undefined: Tangent is undefined where cosine is zero, like at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).

When resolving equations such as \( \tan x + 1 = 0 \) or \( \tan x = -1 \), the unit circle provides solutions through reference angles and understanding of tangent's periodicity. For example, \( \tan x = -1 \) conspicuously occurs at angles \( \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \) while using the unit circle.

Navigating the Unit Circle

The unit circle is a powerful tool in trigonometry, helping us understand and solve trigonometric equations effectively. It is a circle with a radius of one centered at the origin of a coordinate plane. This foundational circle allows the visualization of trigonometric functions, providing the basis for relationships among angles and lengths.Key properties include:

• The x-coordinate of a point on the unit circle corresponds to the cosine value of the angle.
• The y-coordinate corresponds to the sine value of the angle.
• Tangent can be determined as the ratio \( \frac{y}{x} \) of these points.

For example, when solving \( \tan x = -1 \), the unit circle aides by helping identify exact angle positions where this ratio is true. Similarly, using the unit circle, cotangent values are derived as the reciprocal or \( \frac{x}{y} \).Additionally, the unit circle's symmetry proves useful: angles and their reflexive counterparts (e.g., \( \theta \) and \( \pi - \theta \)) are visually understood, allowing easier deduction of equivalent angle solutions within specified intervals like \([0, 2\pi)\). The circle's periodicity naturally aligns with common uses in both pure mathematics and applied sciences.