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Chapter 3: Problem 31

Simplify each expression. $$\frac{\pi}{6}+\frac{\pi}{2}$$

Short Answer

Expert verified
The simplified expression is \(\frac{2\pi}{3}\).

Step by step solution

01

Identify a Common Denominator

To add fractions, they must have the same denominator. The denominators are 6 and 2, and the least common denominator (LCD) is 6. This means we'll need to rewrite \(\frac{\pi}{2}\) to have 6 as the denominator.
02

Rewrite the Second Fraction

Rewrite \(\frac{\pi}{2}\) with the denominator 6. To do this, multiply both the numerator and the denominator by 3: \(\frac{\pi}{2} = \frac{\pi \times 3}{2 \times 3} = \frac{3\pi}{6}\).
03

Add the Fractions

Once the fractions have the same denominator, simply add the numerators: \(\frac{\pi}{6} + \frac{3\pi}{6} = \frac{(\pi + 3\pi)}{6} = \frac{4\pi}{6}\).
04

Simplify the Result

Finally, simplify the fraction \(\frac{4\pi}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{4\pi}{6} = \frac{2\pi}{3}\).

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Key Concepts

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

Least Common Denominator
Adding fractions involves having a common playground for both fractions, which is their denominator. If you have fractions like \(\frac{\pi}{6}\) and \(\frac{\pi}{2}\), you can't add them directly since their denominators are different. The key is to find the Least Common Denominator (LCD), which is the smallest number that both denominators can divide into without leaving a remainder. For the numbers 6 and 2, the LCD is 6.
  • 6 is already the denominator of the first fraction.
  • For the second fraction \(\frac{\pi}{2}\), you need to adjust it to have a denominator of 6.
Finding the LCD ensures that the fractions are on the same "denominator page," making addition straightforward.
Simplifying Fractions
Once fractions are combined, it'is crucial to ensure the resulting fraction is as simple as possible. After adding \(\frac{\pi}{6} + \frac{3\pi}{6}\), you get \(\frac{4\pi}{6}\). Reducing fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator. For \(4\pi\) and 6, the GCD is 2.
  • Divide the numerator by the GCD: \(4\pi\div 2 = 2\pi\)
  • Divide the denominator by the GCD: \(6\div 2 = 3\)
Therefore, the fraction simplifies from \(\frac{4\pi}{6}\) to \(\frac{2\pi}{3}\) giving you a cleaner, simpler expression to work with.
Trigonometry Basics
Trigonometry often involves dealing with fractions and radians, much like our exercise here. In trigonometry, it's common to encounter angles in radian measure, such as \(\pi\), \(\frac{\pi}{2}\), and so forth, because they provide a more natural way of describing angles in terms of the circle.
  • Adding radians, like any other fractions, requires having a common denominator.
  • Using the LCD method is key when handling multiple angles in expressions.
  • Simplifying the resulting fractions helps make the trigonometric calculations easier and cleaner.
Understanding how these basic arithmetic operations relate to trigonometric concepts can significantly aid in solving problems involving angles, circle measurements, and more complex trigonometric calculations.

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