Problem 10 Find the exact value of each exp... [FREE SOLUTION]

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Chapter 8: Problem 10

Find the exact value of each expression. $$ \cot ^{-1} 1 $$

Short Answer

Expert verified

$ \frac{\pi}{4} $

Step by step solution

Understand the problem

We need to find the exact value of $\tan^{-1} 1$. This means finding the angle whose cotangent is 1.

Recall the definition of cotangent

The cotangent of an angle $\theta$ is defined as the reciprocal of its tangent: \[ \cot(\theta) = \frac{1}{\tan(\theta)} \] Hence, we need the angle $\theta$ such that \[\tan(\theta) = 1\] and \[\cot(\theta) = \frac{1}{\tan(\theta)} = 1 \].

Determine the angle

The angle whose tangent is 1 is $\frac{\pi}{4}$ radians, because \[ \tan\left(\frac{\pi}{4}\right) = 1 \] Thus, \[ \cot^{-1}(1) = \frac{\pi}{4} \] radians.

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Key Concepts

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

Cotangent

Cotangent, often abbreviated as cot, is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In other words: \[ \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} \]. This means for an angle \theta, cotangent equals the ratio of the adjacent side to the opposite side in a right-angled triangle. For example, if the tangent of \theta is 1, then: \[ \text{cot}(\theta) = \frac{1}{1} = 1 \]. The cotangent function is very useful in trigonometry, especially when working with inverse functions or solving triangles.

Inverse Functions

Inverse functions reverse the effect of the original function. When we talk about \text{cot}^{-1} (the inverse cotangent), we are looking for the angle whose cotangent is a given value. For example: \[ \text{If } \text{cot}^{-1}(x) = \theta \], then: \[ \text{cot}(\theta) = x \]. In our exercise, \text{cot}^{-1} (1) asks for the angle whose cotangent is 1. This requires us to find the value of $ \theta $ where: \[ \text{tan}(\theta) = \frac{1}{\text{cot} (\theta)} = 1 \].

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the identity are defined. They are useful for simplifying expressions and solving trigonometric equations. Some important identities involving cotangent include:

$ \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} $
$ \text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)} $
$ \text{cot}^2(\theta) + 1 = \text{csc}^2(\theta) $

These identities are crucial in understanding relationships between different trigonometric functions and are often used to transform and solve trigonometric problems.

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91影视

Find the exact value of each expression. $$ \cot ^{-1} 1 $$

Short Answer

Step by step solution

Understand the problem

Recall the definition of cotangent

Determine the angle

Key Concepts

Cotangent

Inverse Functions

Trigonometric Identities

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