/*! 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 61 Use the Law of sines to prove th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 61

Use the Law of sines to prove that all angles in an equilateral triangle must have the same measure.

Short Answer

Expert verified
All angles in an equilateral triangle are \(60^\circ\).

Step by step solution

01

Understanding Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal in length. We denote the sides as \(a\), \(b\), and \(c\); therefore, \(a = b = c\). Our goal is to show that all the angles of the triangle are equal.
02

Applying the Law of Sines

The Law of Sines states that for a triangle with sides \(a\), \(b\), and \(c\), and angles \(A\), \(B\), and \(C\) opposite those sides, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). For an equilateral triangle, we have \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Since \(a = b = c\), we conclude \(\sin A = \sin B = \sin C\).
03

Concluding the Equivalence of Angles

In a triangle, if \(\sin A = \sin B = \sin C\), then \(A = B = C\) because for a specific angle in \(0, \pi)\), the sine function is injective. Thus, in our equilateral triangle, all angles must be equal. Since a triangle's angles sum to \(180^\circ\), each angle must be \(60^\circ\).

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.

Equilateral Triangle
An equilateral triangle is a unique type of triangle where all three sides have equal length. If you picture any triangle, imagine that each side is exactly the same.
This is the hallmark of an equilateral triangle. Because of this equal side length, it has some interesting properties. One of these is that all the angles are also equal.
  • Each angle in an equilateral triangle measures 60 degrees.
  • The sides are usually denoted as \( a = b = c \).
  • This equality of angles and sides makes constructions and calculations involving equilateral triangles both simple and elegant.
Knowing that these characteristics repeat for any equilateral triangle is crucial for understanding geometry and trigonometry principles.
Triangle Angles
Angles are a fundamental concept when it comes to understanding triangles. Each triangle, regardless of type, always adheres to the principle that the sum of its angles equals 180 degrees.
This is an essential geometric truth.
  • In an equilateral triangle, because the sides are equal, the angles must also be equal.
  • If all angles are equal, then each angle must be \( \frac{180^\circ}{3} = 60^\circ \).
Understanding this helps visualize how every triangle's angles relate to each other. For equilateral triangles, this means each corner measures exactly the same. This uniformity in angles is due to the triangle's equal sides, reinforcing each angle's equality.
Recognizing this concept helps keep angle calculations consistent and accurate.
Sine Function
The sine function is an essential tool in trigonometry that relates angles in triangles to side lengths. In any triangle with angles \( A \), \( B \), and \( C \), opposite sides are traditionally denoted as \( a \), \( b \), and \( c \) respectively.
The Law of Sines uses the sine function to establish a relationship between these angles and their opposite sides.
  • The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). It's a profound connection that exists because of how the sine of an angle correlates with the lengths across from it.
  • In an equilateral triangle, this relationship is straightforward since the sides are all equal, meaning \( a = b = c \).
  • Hence, \( \sin A = \sin B = \sin C \), leading us to conclude that \( A = B = C \).
Understanding the sine function and its role in the Law of Sines helps reveal why all angles in an equilateral triangle are equal. It simplifies proving angle equality by showing that equal side lengths naturally enforce equal angles.

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

In an oblique triangle \(A B C, \beta=\frac{2 \pi}{9}, \gamma=\frac{5 \pi}{9},\) and \(a=200 \mathrm{ft} .\) Find the length of \(c .\) Round your answer to the nearest unit.

Determine whether each statement is true or false. The Law of Cosines is a special case of the Pythagorean theorem.

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\sec 270^{\circ}$$

When light passes from one substance to another, such as from air to water, its path bends. This is called refraction and is what is seen in eyeglass lenses, camera lenses, and gems. The rule governing the change in the path is called Snell's law, named after a Dutch astronomer: \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},\) where \(n_{1}\) and \(n_{2}\) are the indices of refraction of the different substances and \(\theta_{1}\) and \(\theta_{2}\) are the respective angles that light makes with a line perpendicular to the surface at the boundary between substances. The figure shows the path of light rays going from air to water. Assume that the index of refraction in air is \(1 .\) (GRAPH CANNOT COPY) If the refraction index for a rhinestone is \(1.9,\) then to what angle is light refracted if it enters the rhinestone at an angle of \(30^{\circ} ?\) Round the answer to two significant digits.

Refer to the following: NASA explores artificial gravity as a way to counter the physiologic effects of extended weightlessness for future space exploration. NASA's centrifuge has a 58 -foot-diameter arm. If two humans are on opposite (red and blue) ends of the centrifuge and their linear speed is 200 miles per hour, how fast is the arm rotating? Express the answer in radians per second to two significant digits.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

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.