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Chapter 3: Problem 8

Find the exact values of the following sums or differences. $$ \frac{\pi}{3}+\frac{\pi}{6} $$

Short Answer

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

Step by step solution

01

Identify the Common Denominator

To subtract the fractions \(\frac{\pi}{6}-\frac{\pi}{4}\), we first need a common denominator. The denominators are 6 and 4. The least common multiple (LCM) of 6 and 4 is 12.
02

Convert Fractions to Common Denominator

Rewrite each fraction with the common denominator of 12: \(\frac{\pi}{6} = \frac{2\pi}{12}\) and \(\frac{\pi}{4} = \frac{3\pi}{12}\).
03

Subtract the Fractions

Now subtract the two fractions with a common denominator: \(\frac{2\pi}{12} - \frac{3\pi}{12} = \frac{(2\pi - 3\pi)}{12} = \frac{-\pi}{12}\)
04

Simplify the Result

The final simplified form is: \(\frac{-\pi}{12}\)

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Key Concepts

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

least common multiple
When dealing with any fraction operation, finding a common denominator is crucial. The least common multiple (LCM) is the smallest number that all denominators can divide into without leaving a remainder.
The LCM helps align different denominators, making operations like addition and subtraction easier.
To find the LCM of two numbers, list the multiples of each number until you find a common one. For example, find the LCM of 6 and 4:
  • Multiples of 6: 6, 12, 18, 24...
  • Multiples of 4: 4, 8, 12, 16...
The first common multiple is 12. Therefore, the LCM of 6 and 4 is 12.
Using the LCM as a common denominator allows us to convert all fractions involved in the operation to the same base, simplifying the process.
fraction subtraction
To subtract fractions, they must first have the same denominator. Let's use the exercise \(\frac{\pi}{6} - \frac{\pi}{4}\). Start by finding the LCM of 6 and 4, which is 12. Convert each fraction to have this common denominator:
\(\frac{\pi}{6} = \frac{2\pi}{12}\) and \(\frac{\pi}{4} = \frac{3\pi}{12}\).
Once the denominators are the same, subtract the numerators: \(\frac{2\pi}{12} - \frac{3\pi}{12} = \frac{(2\pi - 3\pi)}{12} = \frac{-\pi}{12}\).
For fraction subtraction, always ensure denominators match first, then subtract the numerators while keeping the denominator constant. This method keeps calculations straightforward and accurate.
simplifying fractions
After performing operations on fractions, it's essential to simplify the result. A fraction is simplified when the numerator and denominator have no common factors other than 1.
For instance, the result \(\frac{-\pi}{12}\) from our exercise is already in its simplest form.
Here鈥檚 another example: simplify the fraction \(\frac{4}{8}\). Both the numerator and the denominator can be divided by their greatest common divisor (GCD), which is 4. So, \(\frac{4}{8} = \frac{4\div 4}{8\div 4} = \frac{1}{2}\).
Always check each fraction by dividing both parts by their GCD to make sure it's in the simplest form. This step is crucial for keeping equations clean and results easy to read.

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Most popular questions from this chapter

Evaluate each trigonometric function if possible. a. \(\sin (5 \pi / 6)\) b. \(\cos (\pi)\) c. \(\tan (3 \pi / 2)\) d. \(\sec (-\pi / 3)\) e. \(\csc (-\pi / 2)\) f. \(\cot (5 \pi / 4)\)

Suppose that \(\sin \alpha=1 / 4\) and \(\alpha\) is in quadrant II. Use identities to find the exact values of the other five trigonometric functions.

The equation \(f_{1}(x)=f_{2}(x)\) is an identity if and only if the graphs of \(y=f_{1}(x)\) and \(y=f_{2}(x)\) coincide at all values of \(x\) for which both sides are defined. Graph \(y=f_{1}(x)\) and \(y=f_{2}(x)\) on the same screen of your calculator for each of the following equations. From the graphs, make a conjecture as to whether each equation is an identity, then prove your conjecture. $$ \frac{\sin \theta+\cos \theta}{\sin \theta}=1+\cot \theta $$

Simplify each expression by applying the odd/even identities, cofunction identities, and cosine of a sum or difference identities. Do not use a calculator $$ \cos (-3 k) \cos (-k)-\cos (\pi / 2-3 k) \sin (-k) $$

52\. \(\frac{1-\cos ^{2}\left(\frac{x}{2}\right)}{1-\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1-\cos x}{1+\cos x}\)
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