/*! 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 27 Find the lengths of both circula... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 27

Find the lengths of both circular arcs of the unit circle connecting the point (1,0) and the endpoint of the radius corresponding to 3 radians.

Short Answer

Expert verified
The lengths of the shorter and longer arcs connecting the point (1,0) and the endpoint of the radius corresponding to 3 radians in the unit circle are 3 and \(2\pi - 3\), respectively.

Step by step solution

01

Find the angle of the longer arc

To find the angle of the longer arc, we will subtract 3 radians from 2蟺 radians (as it makes a complete circle): Longer arc angle = \(2\pi - 3\)
02

Calculate the lengths of both arcs

To find the lengths of the arcs, we will use the arc length formula: Arc length = radius 脳 angle (in radians) Since it's a unit circle, the radius is 1.
03

Find the length of the shorter arc

We know the angle of the shorter arc is 3 radians and the radius is 1: Shorter arc length = radius 脳 angle = \(1 \times 3 = 3\)
04

Find the length of the longer arc

We already found the angle of the longer arc in Step 1. Now we will multiply it with the radius to find the length of the longer arc: Longer arc length = radius 脳 angle = \(1 \times (2\pi - 3) = 2\pi - 3\) So, the lengths of the shorter and longer arcs are 3 and \(2\pi - 3\), respectively.

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!

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

Find a formula for \(\tan ^{2} \theta\) solely in terms of \(\sin ^{2} \theta\).

Show that $$\sin (\pi-\theta)=\sin \theta$$ for every angle \(\theta\).

Find the lengths of all three sides of a right triangle that has perimeter 29 and has a \(42^{\circ}\) angle.

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\tan \frac{25 \pi}{12}\)

The next two exercises emphasize that \(\sin ^{2} \theta\) does not ?qual \(\sin \left(\theta^{2}\right)\). For \(\theta=-8^{\circ},\) evaluate each of the following: (a) \(\sin ^{2} \theta\) (b) \(\sin \left(\theta^{2}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.