/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 \(y_{1}=\frac{\cos x}{\sin x}, \... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 55

\(y_{1}=\frac{\cos x}{\sin x}, \quad y_{2}=\cot x\)

Short Answer

Expert verified
The functions \(y_{1}\) and \(y_{2}\) are equal.

Step by step solution

01

Simplify the First Function

Start by simplifying the first function. We have \(y_{1}=\frac{\cos x}{\sin x}\). It is already in simplified form.
02

Simplify the Second Function

Now, simplify the second function. We have \(y_{2}=\cot x\). By definition, \(\cot x=\frac{\cos x}{\sin x}\).
03

Compare the Two Functions

Compare the simplified versions of the first and the second function. Since both simplified expressions are \(\frac{\cos x}{\sin x}\), it can be inferred that \(y_{1}\) is indeed equal to \(y_{2}\).

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Key Concepts

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

Cotangent Function
In trigonometry, the cotangent function is a crucial concept. It's one of the basic trigonometric functions and complements sine, cosine, and tangent. The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the opposite side. Mathematically, it's expressed as \( \cot x = \frac{\cos x}{\sin x} \).A few key points about the cotangent function to remember:
  • It's undefined for angles where \( \sin x = 0 \), such as 0, \( \pi \), 2\( \pi \), and so on, since division by zero is not possible.
  • The cotangent function is periodic with a period of \( \pi \). This means its values repeat every \( \pi \) radians.
  • Cotangent is the reciprocal of the tangent function, so \( \cot x = \frac{1}{\tan x} \).
Understanding the cotangent function helps in simplifying and transforming trigonometric expressions for easier manipulation in formulas and equations.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is an essential skill in mathematics. It involves rewriting an expression in a simpler or more convenient form, often using trigonometric identities. This process can simplify calculations and provide deeper insights into the properties of functions. Let's look at the example described in the exercise:
  • The function \( y_{1} = \frac{\cos x}{\sin x} \) is already in a simplified form representing the cotangent function.
  • Rewriting \( \cot x \) as \( \frac{\cos x}{\sin x} \) elucidates its identity and shows that different expressions can represent the same trigonometric function.
By recognizing patterns and using identities, such as \( \sin^2 x + \cos^2 x = 1 \), you can simplify complex trigonometric expressions efficiently. This approach is essential for simplifying equations, solving trigonometric problems, and making calculations more manageable.
Comparison of Functions
Comparing functions in trigonometry is a method to understand their relationships and characteristics. In the provided exercise, both \( y_1 \) and \( y_2 \) are compared to show they are identical functions.This comparison involves:
  • Breaking down each function to see if they represent or simplify to the same expression using definitions and identities.
  • Seeing that \( y_{1} = \frac{\cos x}{\sin x} \) is exactly the same as \( y_{2} = \cot x \).
When learning trigonometry, recognizing that different forms of a function are equivalent is crucial. This understanding allows for seamless transformations between different trigonometric forms, leading to easier problem-solving and a more profound comprehension of mathematical concepts.

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