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Chapter 5: Problem 52

Find the exact value of each expression without using a calculator or table. \(\cot ^{-1}(1)\)

Short Answer

Expert verified
\( \text{cot}^{-1}(1) = \frac{\pi}{4} \)

Step by step solution

01

Understand the Inverse Cotangent Function

The function \(\text{cot}^{-1}(x)\) gives the angle whose cotangent is \(x\). This function is also known as the arccotangent.
02

Identify the Cotangent Value

The problem asks for \( \text{cot}^{-1}(1) \), meaning we need to find the angle \( \theta \) such that \(\text{cot}(\theta) = 1\).
03

Recall Cotangent and Its Standard Angles

The cotangent of an angle \( \theta \) is the ratio of the adjacent side to the opposite side in a right triangle, \(\text{cot}(\theta) = \frac{\text{adjacent}}{\text{opposite}}\). For \( \text{cot}(\theta) = 1 \), the angle must be where the sides are equal, like \(\theta=45^\text{掳}\) or \( \theta=\frac{\pi}{4}\) radians.
04

Identify Appropriate Angle in the Range

Since \( \text{cot}^{-1}(x) \) usually gives a principal value in the range \( (0, \pi) \), we select \( \theta = \frac{\pi}{4} \).

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Key Concepts

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

cotangent
The cotangent function is a fundamental trigonometric function. It is the reciprocal of the tangent function, which means \(\text{cot}(\theta) = \frac{1}{\text{tan}(\theta)}\). Additionally, the cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side: \(\text{cot}(\theta) = \frac{\text{adjacent}}{\text{opposite}}\).

Here are some key points to remember about the cotangent function:
  • The cotangent function is not defined for angles where the tangent function is zero because division by zero is undefined.
  • Typical values and angles include \(\text{cot}(45^\text{掳}) = 1\) and \(\text{cot}(135^\text{掳}) = -1\).

Understanding these properties and values will help us find the exact value of expressions involving the cotangent function.
arccotangent
The arccotangent, also known as the inverse cotangent function, is denoted as \(\text{cot}^{-1}(x) \). This function returns the angle whose cotangent is x.

Essentially, if you know the value of \(\text{cot}(\theta)\), you can use \(\text{cot}^{-1}(x)\) to find the angle \(\theta\). Here are a few important points to keep in mind:
  • The range of \(\text{cot}^{-1}(x)\) is typically chosen to be between 0 and \(\pi\) (exclusive).
  • The function \(\text{cot}^{-1}(x)\) will yield an acute angle in this range.

To use the arccotangent to find an angle, you should be familiar with standard values from the unit circle. For instance, \(\text{cot}^{-1}(1)\) results in \(\frac{\pi}{4} \) radians, or 45 degrees, because \( \text{cot} \left( \frac{\pi}{4} \right ) = 1\).

In summary, the arccotangent helps us determine the angle when we know the cotangent value.
exact value of trigonometric expressions
Finding the exact value of trigonometric expressions involves determining precise values without the use of a calculator or table. This requires an understanding of the properties and key angles commonly found in trigonometric functions.

For example, to find the exact value of \(\text{cot}^{-1}(1)\), you can follow these steps:
  • Identify that \(\text{cot}^{-1}(x)\) asks for an angle where the cotangent equals x.
  • Recognize that for \(\text{cot}^{-1}(1)\), you need the angle where \(\text{cot}(\theta) = 1\).
  • Recall from your knowledge of standard angles that \(\frac{\pi}{4} \) radians (or 45 degrees) is an angle where cotangent is 1.

Thus, the exact value is \(\frac{\pi}{4}\).

Using standard angles and known values from the unit circle helps us find the exact values quickly and accurately. This method enables you to solve trigonometric problems without relying on additional tools, which is particularly helpful in exams and deeper understanding of the subject.

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