Problem 6 - Precalculus: A Concise Course

Question

Fill in the blanks. The domain of \(y=\cot x\) is all real numbers such that ______.

Step-by-Step Solution

Answer

The domain of \(y=\cot x\) is all real numbers such that \(x ≠ n\pi\), where \(n\) is an integer.

Step 1: Understand the cotangent function

The cotangent function, \(\cot x\), is the reciprocal of the tangent function, expressed as \(\cot x = \frac{1}{\tan x}\). Due to this relationship, whenever \(\tan x\) equals zero, \(\cot x\) is undefined.

Step 2: Find the zeros of tangent function

The tangent function equals zero whenever \(x = n\pi\). This is because the tangent function is the ratio of the sine to the cosine, and this ratio equals zero whenever the numerator (sine) equals zero. Sine equals zero at \(n\pi\), where \(n\) is an integer.

Step 3: Define the domain of the cotangent function

Based on which values of \(x\) lead the cotangent function to be undefined, the domain of \(y=\cot x\) would be all real numbers except those for which \(x = n\pi\), where \(n\) is an integer.

Key Concepts

Reciprocal Trigonometric Functions

The study of reciprocal trigonometric functions offers a deeper understanding of trigonometry by exploring the inherent relationships between these functions. A reciprocal trigonometric function essentially inverts another trigonometric function's value. For instance, the cotangent function is the reciprocal of the tangent function, and is expressed as \( \cot x = \frac{1}{\tan x} \). It's important to remember that since tangent is \(\sin x / \cos x\), cotangent can also be represented as \(\cos x / \sin x\).

Interestingly, this reciprocal relationship means that wherever the tangent function has a value, the cotangent function has a reciprocal value, unless the tangent function's value is zero. In that case, because division by zero is undefined in mathematics, the cotangent function is also undefined. This relationship is key to understanding the behavior of cotangent and its domain. By examining reciprocal functions, students can gain insight into how these functions mirror and depend on each other.

Zeros of Tangent Function

Identifying the zeros of the tangent function is a fundamental aspect of trigonometry that influences the domain of its reciprocal, the cotangent function. Tangent zeros occur where the sine function—which represents the numerator in the \(\tan x = \frac{\sin x}{\cos x}\) ratio—equates to zero. These points are found at \(x = n\pi\), for every integer value of \(n\), because \(\pi\) radians represents 180 degrees, or a half turn on the unit circle, where the sine wave intersects the horizontal axis.

Understanding these zeros is crucial because at these exact points, the cotangent function, as the reciprocal, will be undefined due to the impossibility of dividing by zero. It's these zeros of the tangent function that essentially carve out the 'holes' in the domain of the cotangent function. Thus, grasping where and why these zeros occur is vital for solving problems related to the domain of reciprocal trigonometric functions.

Trigonometry

Trigonometry is an area of mathematics that studies the relationships between the angles and sides of triangles, especially right-angled triangles. At the heart of trigonometry are six functions—sine (sin), cosine (cos), tangent (tan), and their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). These functions are essential in various fields, including physics, engineering, and even music theory.

In the context of our problem, trigonometry helps us describe circular motion and oscillations via the unit circle—a circle with a radius of one. The cotangent function, being a part of trigonometry, is associated with angles created in the unit circle and their corresponding points. Every trigonometric function, including cotangent, has specific input values, known as its domain, over which it is defined. The cotangent function in particular has breaks in its curve at regular intervals—these intervals are directly linked to the zeros of its counterpart, the tangent function, marking points where cotangent is undefined and thus shaping its domain.