Problem 110 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cot \theta=-6$$

Step-by-Step Solution

Answer

The statement is possible since cotangent can take any real value.

Step 1: Understand the Cotangent Function

The cotangent function is the reciprocal of the tangent. It is defined as: \[ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \] Cotangent is undefined where tangent is zero, but it is possible anywhere else on its domain.

Step 2: Determine the Range of Cotangent Function

The range of the cotangent function is the set of all real numbers, \((-\infty, \infty)\). This means cotangent can take any real number value.

Step 3: Analyze the Given Value

Since \(\cot \theta = -6\) is a real number and cotangent can have any real number value due to its range, this value is possible for some angle \(\theta\).

Step 4: Provide Conclusion Based on Analysis

Since \(\cot \theta = -6\) is a real number, and the range of cotangent includes all real numbers, it is indeed possible for \(\theta\) to exist such that \(\cot \theta = -6\).

Key Concepts

Cotangent Function

The cotangent function is one of the six core trigonometric functions. It is defined as the reciprocal of the tangent function. This means it is expressed as:

• \[ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \]

Cotangent is mainly useful in trigonometry for understanding different properties and behaviors of angles, especially when dealing with right triangles.
However, the cotangent function does have some points where it is not defined. This occurs when the tangent function itself is zero, because division by zero is undefined in mathematics. Whenever this happens, it typically refers to angles where the sine is zero, such as at multiples of \( \pi \).
Therefore, while cotangent is undefined at these points, it generally exists for other values of \( \theta \), making it a versatile function in trigonometry.

Angle Analysis

Analyzing angles often involves associating the angle with its corresponding trigonometric function value. For the cotangent function, we determine the existence of a specific angle \( \theta \) by analyzing whether a particular cotangent result can correspond to a real angle.
For instance, consider the statement \( \cot \theta = -6 \). The question is whether there is an angle \( \theta \) for which this is true.

• Since cotangent values can be negative, it implies the reference angle can lie in the second or fourth quadrants of the unit circle, where cosine and sine ratios result in a negative cotangent.
• Furthermore, since \( \cot \theta \) deals with the ratio of \( \cos \theta \) to \( \sin \theta \), any real number can be achieved, given that the range includes negative numbers, positive numbers, and the behavior of radian measures.

Hence, the angle analysis confirms that a negative value like -6 for cotangent is feasible, meaning an angle \( \theta \) can indeed exist that fulfills this condition.

Range of Trigonometric Functions

Understanding the range of trigonometric functions helps us know what outputs we can expect from them. For the cotangent function, the range is incredibly broad.

• The range of the cotangent function is \( (-\infty, \infty) \). This means cotangent can produce any real number value.
• This comprehensive range arises because the cotangent function is based on division of cosine by sine, which can trend towards any real value except where terms are undefined.

Know that for other trigonometric functions, the range is more limited. For example, the sine and cosine functions are bounded between -1 and 1, while the tangent function spans from negative to positive infinity, except where undefined.
Ultimately, understanding the range of the cotangent function enables us to affirm any real number input result, such as -6, is achievable for some angle \( \theta \). This makes cotangent a function that is deeply dependable when solving trigonometric equations.