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Chapter 3: Problem 11

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

Identify the fractions

Recognize the two fractions involved in the expression: \ \ \ \ \ \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\).
02

Find a common denominator

To subtract the fractions, a common denominator is needed. The denominators are 4 and 3. The least common multiple of both numbers is 12. Therefore, we rewrite the fractions with a common denominator of 12: \ \ \ \ \ - \(\frac{\pi}{4} = \frac{3\pi}{12}\) and \(\frac{\pi}{3} = \frac{4\pi}{12}\).
03

Perform the subtraction

Subtract the fractions: \ \ \ \ \[ \frac{3\pi}{12} - \frac{4\pi}{12} = \frac{3\pi - 4\pi}{12} = \frac{-\pi}{12} \]
04

Simplify the expression

The fraction simplifies directly to: \ \ \ \ \(\frac{-\pi}{12}\)

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Key Concepts

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

least common multiple
When subtracting fractions, you often need the least common multiple (LCM). The LCM of two numbers is the smallest number that both can divide without leaving a remainder. For example, for the numbers 4 and 3, the LCM is 12 because both 4 and 3 can divide it evenly. Finding the LCM helps in making the denominators of fractions the same, making subtraction or addition possible.
finding common denominators
To subtract fractions, you must first have a common denominator. To get this, you often find the LCM of the denominators. Let's consider the original exercise: \(\frac{\pi}{4} - \frac{\pi}{3}\). To subtract, the fractions need the same bottom number. We found that the LCM of 4 and 3 is 12. So, we can rewrite \(\frac{\pi}{4}\) as \(\frac{3\pi}{12}\) and \(\frac{\pi}{3}\) as \(\frac{4\pi}{12}\). Now, they have a common denominator making subtraction easy.
simplifying fractions
Simplifying fractions helps you to make them easier to understand and work with. After we did the subtraction, \(\frac{3\pi}{12} - \frac{4\pi}{12} = \frac{-\pi}{12}\). This fraction is already in its simplest form. Simplifying means to reduce the numerator and the denominator to their smallest values. Always check if you can simplify further by dividing both the top and bottom by their Greatest Common Divisor (GCD).
trigonometric functions
While the given exercise does not directly involve trigonometric functions, having a foundation in fractions is crucial in trigonometry. Many trigonometric identities and equations require adding or subtracting fractions. Trigonometric functions like sine, cosine, and tangent involve angles that may be expressed in terms of \(\pi\). Understanding how to manipulate fractions will make working with these functions much simpler.

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Most popular questions from this chapter

Find the exact value of \(\sin (x / 2)\) given that \(\cos (x)=-1 / 4\) and \(\pi / 2

Find the exact value of \(\cos (\alpha+\beta)\) if \(\sin \alpha=-7 / 25\) and \(\sin \beta=8 / 17,\) with \(\alpha\) in quadrant IV and \(\beta\) in quadrant II.

Find the point that lies midway between \((\pi / 3,1)\) and \((\pi / 2,1)\)

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. $$ \cot x+\sin x=\frac{1+\cos x-\cos ^{2} x}{\sin x} $$

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