/*! 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 5 Find the exact values of the fol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 5

Find the exact values of the following sums or differences. $$ \pi+\frac{\pi}{2} $$

Short Answer

Expert verified
\( \frac{\text{\pi}}{3} \)

Step by step solution

01

Convert the angles to a common denominator

In order to add the fractions, they need to have a common denominator. The denominators are 4 and 12. The least common denominator of 4 and 12 is 12. Thus, convert \( \frac{\frac{\text{\text{pi}}}{4}} \) to have a denominator of 12.
02

Adjust the first fraction

Convert \( \frac{\text{\text{pi}}}{4} \) to a fraction with denominator 12: \( \frac{\text{\text{pi}}}{4} = \frac{3\text{\text{pi}}}{12} \).
03

Add the fractions

Now add the fractions: \( \frac{3\text{\text{pi}}}{12} + \frac{\text{\text{pi}}}{12} = \frac{4\text{\text{pi}}}{12} \).
04

Simplify the result

Simplify \( \frac{4\text{\text{pi}}}{12} \) by dividing both the numerator and the denominator by 4: \( \frac{4\text{\text{pi}}}{12} = \frac{\text{\text{pi}}}{3} \).

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 you need to add or subtract fractions, they must share a common denominator.
This means that both fractions need the same number on the bottom part.
For example, in this exercise, we had \(\frac{\text{\pi}}{4}\) and \(\frac{\text{\pi}}{12}\).
Since their denominators are different (4 and 12), we need to convert them to the same denominator.
The least common denominator of 4 and 12 is 12.
Therefore, we convert \(\frac{\text{\pi}}{4}\) to \(\frac{3\text{\pi}}{12}\) because \(\frac{\text{\pi}}{4} = \frac{3\text{\pi}}{12}\).
Now, both fractions have the same denominator, allowing us to proceed with addition.
Angle Addition
Angle addition in trigonometry often involves summing angles expressed in radians.
When the angles are given as fractions of \text{\pi}\, you might need to bring them to a common denominator first.
Once both fractions share the same denominator, you can add the numerators directly while keeping the denominator the same.
In our example, after converting \(\frac{\text{\pi}}{4}\) to \(\frac{3\text{\pi}}{12}\), we simply add \(\frac{3\text{\pi}}{12}\) and \(\frac{\text{\pi}}{12}\).
The result of this addition is \(\frac{4\text{\pi}}{12}\).
Simplifying Fractions
Simplifying fractions is about reducing them to their smallest form.
This often means dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, \(\frac{4\text{\pi}}{12}\) can be simplified by dividing both 4 and 12 by their GCD, which is 4.
\(\frac{4\text{\pi}}{12}\) divided by 4 becomes \(\frac{\text{\pi}}{3}\).
Simplifying fractions makes your final answer cleaner and easier to understand.

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

Match each expression with an equivalent expression from \((a)-(h)\) Do not use a calculator: a. \(\cos (0)\) b. \(-\cos \left(44^{\circ}\right)\) c. \(-\tan \left(44^{\circ}\right)\) d. \(\cot \left(\frac{5 \pi}{14}\right)\) e. \(-\cos \left(46^{\circ}\right)\) f. \(\csc \left(\frac{\pi-2}{2}\right)\) g. \(\sin \left(46^{\circ}\right)\) h. \(\sin \left(44^{\circ}\right)\) $$ \cos \left(44^{\circ}\right) $$

Prove that each of the following equations is an identity. HINT \(\ln (a / b)=\ln (a)-\ln (b)\) and \(\ln (a b)=\ln (a)+\ln (b)\) for \(a>0\) and \(b>0\) $$ \ln |\sec \alpha+\tan \alpha|=-\ln |\sec \alpha-\tan \alpha| $$

Find the exact value of \(\tan (x / 2)\) given that \(\sin (x)=-\sqrt{8 / 9}\) and \(\pi

Match each expression with an equivalent expression from \((a)-(h)\) Do not use a calculator: a. \(\cos (0)\) b. \(-\cos \left(44^{\circ}\right)\) c. \(-\tan \left(44^{\circ}\right)\) d. \(\cot \left(\frac{5 \pi}{14}\right)\) e. \(-\cos \left(46^{\circ}\right)\) f. \(\csc \left(\frac{\pi-2}{2}\right)\) g. \(\sin \left(46^{\circ}\right)\) h. \(\sin \left(44^{\circ}\right)\) $$ \cot \left(134^{\circ}\right) $$

Solve each problem. Find the exact value of \(\tan (2 \alpha)\) given that \(\sin (\alpha)=-4 / 5\) and \(\alpha\) is in quadrant III.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.