Problem 82 - A Graphical Approach to Precalculus with Limits

Question

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\cot \left(n \cdot 180^{\circ}\right)$$

Step-by-Step Solution

Answer

Undefined, because \( \cot(n \cdot 180^\circ) = \frac{(-1)^n}{0} \).

Step 1: Analyze the Problem

We need to determine the value of \( \cot(n \cdot 180^\circ) \). The cotangent function, \( \cot(\theta) \), is defined as \( \frac{\cos(\theta)}{\sin(\theta)} \). Our goal is to see how \( \cot(n \cdot 180^\circ) \) behaves, knowing \( n \cdot 180^\circ \) is an integer multiple of 180 degrees.

Step 2: Evaluate Sine and Cosine at \(n \times 180^\circ\)

At \( n \cdot 180^\circ \), we know that \( \sin(n \cdot 180^\circ) = 0 \) because sine of any multiple of \(180^\circ\) is 0. Simultaneously, \( \cos(n \cdot 180^\circ) = (-1)^n \). These will help in evaluating the \( \cot \) function.

Step 3: Calculate the Cotangent

Using the definitions in the previous step, the cotangent becomes \( \cot(n \cdot 180^\circ) = \frac{\cos(n \cdot 180^\circ)}{\sin(n \cdot 180^\circ)} = \frac{(-1)^n}{0} \). This division by zero renders the expression undefined.

Key Concepts

Cotangent

The cotangent function, often abbreviated as "cot," is a basic trigonometric function that is derived from the tangent function. Specifically, cotangent is the reciprocal of the tangent, often expressed as:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

As you can see, cotangent relies on the values of cosine and sine. If at any point sine becomes zero, the cotangent ratio becomes undefined because division by zero is not allowed. It’s important to remember this relationship, as it helps when dealing with various trigonometric problems, especially those involving angles that are integer multiples of 180 degrees.

Integer Multiples

When we talk about integer multiples in trigonometry, we often refer to angles that are multiplied by a whole number. A clear example is when we take a basic angle, like 180 degrees, and an integer \(n\) to form angles like \(n \cdot 180^{\circ}\).

• For positive \(n\), these angles rotate around the unit circle multiple times, returning to the starting point.
• For negative \(n\), they do the same, but in the opposite direction.

Understanding integer multiples is crucial in trigonometry because they allow us to explore patterns and symmetries in the trigonometric functions, affecting their periodic nature and causing functions like sine and cosine to repeat their values.

180 Degrees

The angle of 180 degrees plays a critical role in geometry and trigonometry. It represents a straight line and half a complete revolution around a circle. When dealing with multiple angles, such as \(n \cdot 180^{\circ}\), different trigonometric functions exhibit specific behaviors:

• \( \sin(n \cdot 180^{\circ}) = 0 \): Sine returns to zero as it completes half a revolution each time.
• \( \cos(n \cdot 180^{\circ}) = (-1)^n \): Cosine alternates between 1 and -1 as n changes from even to odd numbers.

This understanding informs the analysis of trigonometric functions like cotangent, indicating which values or situations would lead to undefined expressions.

Undefined Expressions

In mathematics, an expression is deemed undefined if it results in a value that does not make sense or cannot be computed using standard arithmetic operations. A common cause for undefined expressions is division by zero, as occurs with the cotangent function.

• When \( \sin(\theta) = 0 \), any expression involving \( \frac{1}{\sin(\theta)} \) becomes undefined.
• In the case of \( \cot(n \cdot 180^{\circ}) = \frac{(-1)^n}{0} \), this directly results in an undefined expression because sine of any integer multiple of 180 degrees is zero, leading to division by zero.

Handling undefined expressions in trigonometry requires care, as they signal angles or conditions where the functions do not produce standard numerical outputs.