/*! 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 3 Find the exact values of the fol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 3

Find the exact values of the following sums or differences. \(\frac{\pi}{4}-\frac{\pi}{3}\)

Short Answer

Expert verified
\frac{-\pi}{12}

Step by step solution

01

Understand the Problem

Identify the given values and the operation required. We need to subtract two fractions of \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\).
02

Find the Common Denominator

Determine the least common denominator (LCD) for the fractions. The denominators are 4 and 3, so the LCD is 12.
03

Convert Fractions to Common Denominator

Rewrite each fraction with the common denominator. \(\frac{\pi}{4} = \frac{3\pi}{12}\) and \(\frac{\pi}{3} = \frac{4\pi}{12}\).
04

Perform the Subtraction

Subtract the numerators of the fractions with the common denominator. \(\frac{3\pi}{12} - \frac{4\pi}{12} = \frac{3\pi - 4\pi}{12} = \frac{-\pi}{12}\).
05

Simplify the Answer

The result is already simplified. The exact value of the subtraction is \(\frac{-\pi}{12}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Common Denominator
When dealing with fractions, it's not always straightforward to subtract or add them. This is especially true when the fractions have different denominators. The key to solving such problems is finding a common denominator.
The common denominator is essentially a shared multiple of the two denominators of the given fractions. In this exercise, we had to find a common denominator for 4 and 3. The smallest common multiple of these numbers is 12. Once both fractions are rewritten with this common denominator, the problem becomes a lot simpler.
Fraction Subtraction
Subtracting fractions might feel intimidating, but it's quite manageable once a common denominator is found.
Here鈥檚 the step-by-step process once you have the common denominator:
  • Convert each fraction to have this common denominator.
  • Subtract the numerators while keeping the common denominator the same.
  • Simplify the resultant fraction if possible.
In our example, we converted \( \frac{\pi}{4} \) to \( \frac{3\pi}{12} \) and \( \frac{\pi}{3} \) to \( \frac{4\pi}{12} \). Then, we performed \( \frac{3\pi}{12} - \frac{4\pi}{12} \), which gave \( \frac{-\pi}{12} \).
Exact Values in Trigonometry
Exact values refer to the most simplified forms of results, especially in trigonometric calculations. These values are usually represented without decimal approximations.
In this exercise, since we used \( \pi \) which is crucial in trigonometry, the result \( \frac{-\pi}{12} \) is an exact value because it involves \( \pi \).
These exact forms are important in higher-level mathematics and physics because they maintain accuracy and are universally recognized.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Derive the identity \(\cos (2 x)=\cos ^{2} x-\sin ^{2} x\) from a product- to-sum identity.

Prove that each equation is an identity. \(\frac{\sin 4 t}{4}=\cos ^{3} t \sin t-\sin ^{3} t \cos t\)

Find the period and equations of the asymptotes for the function \(y=\cot (2 x)+1\)

Complete the following odd and even identities. a. \(\sin (-x)=\) ____ b. \(\cos (-x)=\)____ c. \(\tan (-x)=\)____ d. \(\csc (-x)=\)____ e. \(\sec (-x)=\)____ f. \(\cot (-x)=\)____

Write each expression as a function of \(\alpha\) alone. $$ \cos (2 \pi-\alpha) $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.