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Chapter 6: Problem 21

Find the following sums or differences in terms of \(\pi .\) $$\frac{\pi}{4}+\frac{\pi}{3}$$

Short Answer

Expert verified
\( \frac{7\pi}{12} \)

Step by step solution

01

Identify the Denominators

The given fractions are \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \). The denominators are 4 and 3, respectively.
02

Find the Least Common Denominator (LCD)

To add the fractions, find the least common denominator (LCD) of 4 and 3. The LCD is 12.
03

Convert to Equivalent Fractions

Convert \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \) to fractions with the common denominator 12. \(\frac{\pi}{4} = \frac{3\pi}{12}\) and \(\frac{\pi}{3} = \frac{4\pi}{12}\).
04

Add the Fractions

Add the fractions with the common denominator: \( \frac{3\pi}{12} + \frac{4\pi}{12} = \frac{3\pi + 4\pi}{12} = \frac{7\pi}{12}\).

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Key Concepts

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

least common denominator
When adding fractions, it's essential to find the least common denominator (LCD). The LCD is the smallest number that both denominators divide into evenly.
For example, let's add \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\). The denominators are 4 and 3.
To find the LCD:
  • List the multiples of each denominator
  • Identify the smallest common multiple

  1. Multiples of 4: 4, 8, 12, 16...
  2. Multiples of 3: 3, 6, 9, 12...
The smallest common multiple is 12, so the LCD is 12.
The LCD makes adding fractions with different denominators possible by converting them to have the same denominator.
equivalent fractions
Equivalents fractions have different numerators and denominators but represent the same value. Converting fractions to have a common denominator involves creating equivalent fractions.
Using our fractions \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\) and the LCD of 12:
  • Convert \(\frac{\pi}{4}\): Multiply numerator and denominator by 3 => \(\frac{\pi \cdot 3}{4 \cdot 3} = \frac{3\pi}{12}\)
  • Convert \(\frac{\pi}{3}\): Multiply numerator and denominator by 4 => \(\frac{\pi \cdot 4}{3 \cdot 4} = \frac{4\pi}{12}\)
Now, both fractions have a common denominator of 12, making addition straightforward.
sum of fractions
Adding fractions becomes simple once they share a common denominator. Using the equivalent fractions we obtained:
  • \(\frac{3\pi}{12}\)
  • \(\frac{4\pi}{12}\)
To add these fractions:
  • Add the numerators: \(3\pi + 4\pi = 7\pi\)
  • Keep the common denominator: 12
The result is \(\frac{7\pi}{12}\).
Here鈥檚 the general process illustrated:
  1. Find the LCD
  2. Convert to equivalent fractions
  3. Add the numerators
  4. Keep the common denominator
In this case, \(\frac{\pi}{4} + \frac{\pi}{3} = \frac{7\pi}{12}\).

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Most popular questions from this chapter

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