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Chapter 1: Problem 24

Evaluate the following without a calculator. Some of these expressions are undefined. $$ \cot (-\pi) $$

Short Answer

Expert verified
The expression is undefined because \( \text{sin}( \pi ) = 0 \).

Step by step solution

01

Understand the Cotangent Function

The cotangent function, \(\text{cot}( \theta )\), is defined as \[ \text{cot}( \theta ) = \frac{\text{cos}( \theta )}{\text{sin}( \theta )} \].
02

Determine the Angles

Evaluate \( \text{cot}(- \pi) \). First note that \( - \pi \) is an angle, which is the same as \( \pi \) in the standard position because the cotangent of an angle is periodic with period \( \pi \).
03

Evaluate Trigonometric Functions

Calculate \[ \text{cos}(- \pi) = \text{cos}(\pi) = -1 \] and \[ \text{sin}(- \pi) = \text{sin}( \pi) = 0 \].
04

Determine if the Expression is Defined

Since \(\text{sin}( \pi ) = 0\), the expression \(\text{cot}( -\pi ) = \frac{\text{cos}( -\pi )}{\text{sin}( -\pi )} = \frac{-1}{0}\) is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent Function
The cotangent function, denoted as \(\text{cot}(\theta)\), is a fundamental trigonometric function. It is defined as the ratio of the cosine to the sine of the given angle \(\theta\). Mathematically, it is expressed as: \[\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}\].
This means that the cotangent of an angle gives us the precision between the x-coordinate and y-coordinate on the unit circle for that specific angle.
Here are some key points about the cotangent function:
  • It is periodic with a period of \(\text{蟺}\). This means \(\text{cot}(\theta + n\text{蟺}) = \text{cot}(\theta)\) for any integer \(n\).
  • It is undefined whenever the sine of the angle \(\theta\) is zero.
Knowing these properties helps us quickly evaluate or determine if a cotangent function is defined or not for a given angle.
Evaluating Trigonometric Expressions
To evaluate trigonometric expressions like \(\text{cot}(-\text{蟺})\), follow these steps:
  • Identify the angle and look for known values of sine and cosine at this angle.
  • Use the unit circle to find coordinates corresponding to the angle.
  • Apply the definition of the cotangent function: \(\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}\).
For \(\text{cot}(-\text{蟺})\), we start by understanding that \(-\text{蟺}\) is an angle. By the periodic nature of cotangent, \(\text{cot}(-\text{蟺})\) is the same as \(\text{cot}(\text{蟺})\).
On the unit circle, at angle \(\text{蟺}\), we have:
\[\text{cos}(\text{蟺}) = -1\] and \[\text{sin}(\text{蟺}) = 0\].
Thus, \[\text{cot}(\text{蟺}) = \frac{-1}{0}\], which simplifies the result.
Undefined Trigonometric Values
Trigonometric expressions can be undefined when the denominator of a fraction is zero. For the cotangent function, this happens when the sine of the angle is zero since \(\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}\).
Understanding and identifying when sinusoidal values are zero is crucial to determining when expressions like \(\text{cot}(-\text{蟺})\) become undefined. In our current problem:
  • We have \(\text{cos}(\text{蟺}) = -1\) and \(\text{sin}(\text{蟺}) = 0\).
  • Thus, our denominator is zero, rendering \(\text{cot}(-\text{蟺})\) undefined.
This is akin to understanding vertical asymptotes in the graph of cotangent, where the function approaches infinity or negative infinity.
Recognizing where these undefined points lie is crucial in properly evaluating trigonometric expressions.

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