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Chapter 1: Problem 1

Give the reciprocal of each number. $$ 7 $$

Short Answer

Expert verified
The reciprocal of 7 is \( \frac{1}{7} \).

Step by step solution

01

Understanding the Concept of a Reciprocal

A reciprocal of a number is 1 divided by that number. For a nonzero number \( x \), the reciprocal is \( \frac{1}{x} \). If we multiply a number by its reciprocal, the result is always 1.
02

Identifying the Given Number

In this problem, the given number is 7. We are tasked with finding the reciprocal of 7.
03

Calculating the Reciprocal

To find the reciprocal of 7, divide 1 by 7. This is expressed as \( \frac{1}{7} \). Thus, the reciprocal of 7 is \( \frac{1}{7} \).
04

Verification

To verify the calculation, multiply the number 7 by its reciprocal. Compute \( 7 \times \frac{1}{7} = 1 \). Since the product is 1, our reciprocal is correct.

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Key Concepts

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

Trigonometry Basics
Trigonometry might seem like it's only about triangles and angles, but its concepts are used widely across various mathematical operations, including finding reciprocals.
The idea of a reciprocal connects closely with trigonometric functions.
  • In trigonometry, each of the sine, cosine, and tangent functions has a reciprocal function: cosecant, secant, and cotangent respectively.
  • For example, if you know the sine of an angle is \(\sin(\theta)=a\), its reciprocal, cosecant, is \(\csc(\theta)=\frac{1}{a}\).
  • The reciprocal relationships help in solving a variety of trigonometric equations.
In many ways, understanding basic reciprocal relationships supports learning more advanced trigonometric concepts.
Number Operations
Number operations are the building blocks of mathematics and play a vital role in understanding reciprocals. Here's a closer look:
  • To find the reciprocal of a number, you perform the operation of division.
  • Dividing 1 by any nonzero number gives you its reciprocal.
  • For example, the reciprocal of 7 is found by calculating \(\frac{1}{7}\).
These operations are simple yet powerful. They allow us to reverse the concept of multiplication, helping solve equations and understand the behavior of fractions.
Arithmetical Concepts
Arithmetic is the cornerstone of all branches of mathematics, and understanding its concepts is essential. Let's delve into how they relate to reciprocals:
  • The concept of a reciprocal belongs to the family of fractions.
  • A reciprocal is essentially flipping or inverting a fraction.
    • This means the numerator becomes the denominator and vice versa.
    • For instance, the reciprocal of a positive integer like 7 becomes \(\frac{1}{7}\).
  • Multiplying a number by its reciprocal, such as 7 and its reciprocal \(\frac{1}{7}\), yields 1, exemplifying the inverse nature of reciprocals.
Mastering these concepts will make tackling arithmetic problems simpler and improve comprehension of more advanced mathematical principles.

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Most popular questions from this chapter

Write each of the following in terms of \(\sin \theta\) and \(\cos \theta\); then simplify if possible: \(\sec \theta \cot \theta\)

Show that each of the following statements is an identity by transforming the left side of each one into the right side. \(\sec \theta-\cos \theta=\frac{\sin ^{2} \theta}{\cos \theta}\)

Add and subtract as indicated. Then simplify your answers if possible. Leave all answers in terms of \(\sin \theta\) and/or \(\cos \theta\). $$ \frac{1}{\cos \theta}-\cos \theta $$

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \cos \theta \tan \theta=\sin \theta $$

Simplify the expression \(\sqrt{9 x^{2}-81}\) as much as possible after substituting \(3 \sec \theta\) for \(x\).
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