/*! 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 53 Without using symbols, state the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 53

Without using symbols, state the Law of cosines in your own words.

Short Answer

Expert verified
The Law of Cosines states that for any triangle (not just right-angled), the square of the length of one side equals the sum of the squares of the lengths of the other two sides, minus two times the product of these two sides lengths and the cosine of the included angle.

Step by step solution

01

Understanding the Law of Cosines

The Law of Cosines is a rule in trigonometry that helps calculate the length of an unknown side in a triangle. Decoding it in words, it goes like this: 鈥淟et鈥檚 think of a triangle that has three sides and one of the angles is not a right angle. The square of the length of the side opposite this angle equals the sum of the squares of the lengths of the other two sides, and from this, we subtract two times the product of these two sides lengths and the cosine of the non-right angle.
02

Breaking Down the Components

Let's break down the rule:1. 'The square of the length of the side opposite the angle' means taking the side across from the angle in question and squaring its length.2. 'The sum of the squares of the lengths of the other two sides' implies adding the squares of the lengths of the other two sides of the triangle.3. 'Subtract two times the product of these two sides lengths and the cosine of the non-right angle' means subtracting twice the multiplication of the lengths of the two sides times the cosine of the given angle.

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.

Trigonometry
Trigonometry is an essential branch of mathematics that deals with the relationships between the sides and angles of triangles. It is particularly useful in solving problems involving non-right triangles, where techniques like the Law of Cosines come into play.
  • Trigonometry is widely used in various fields such as physics, engineering, astronomy, and even art.
  • Each angle in a triangle has a direct relationship with its opposite side, and trigonometric functions like sine, cosine, and tangent help express these relationships.
  • Understanding trigonometric concepts begins with mastering how these functions relate angles to side lengths.
By understanding the basic principles of trigonometry, students can easily apply them to more complex scenarios, such as calculating distances and angles in real-world situations.
Triangle Sides
Triangles, one of the fundamental shapes in geometry, consist of three sides. Understanding the properties of these sides is crucial to solving problems involving triangles.

  • Each triangle is defined by three sides and three angles, and they come in various types: scalene, isosceles, equilateral, and right triangles, depending on their side length and angles.
  • The relationship between the sides of a triangle is critical because it helps in identifying unknown side lengths. Learn how these relationships work with different theorems, such as the Law of Sines and the Law of Cosines.
  • It's important to note that the sum of the angles within any triangle is always \(180^{\circ}\).
In particular, the Law of Cosines is a valuable tool here. It provides a way to calculate any side when we know the lengths of two sides and the measure of the non-included angle. This makes finding unknown side lengths straightforward when working with non-right triangles.
Non-Right Angle
A non-right angle in a triangle is any angle that is neither a \(90^{\circ}\) angle nor defines a right triangle. Many real-world triangles consist of angles that are non-right, making it essential to understand how to work with them.

  • In triangles with a non-right angle, traditional right triangle rules do not apply fully. Instead, we use laws and theorems such as the Law of Cosines.
  • These techniques let us solve for unknown measurements without the requirement of having a right angle.
  • The Law of Cosines, in particular, lends itself well to problems involving non-right triangles, since it relates all three sides and a particular included angle, allowing us to find missing information.
By familiarizing ourselves with these concepts, interpreting and solving problems involving non-right angles become straightforward even for students new to trigonometry.

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

Solve: \(2 \sin ^{2} x-1=0,0 \leq x<2 \pi\)

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$\left(4,90^{\circ}\right)$$

If \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j},\) find a vector orthogonal to \(\mathbf{v}\)

The wind is blowing at 10 knots. Sailboat racers look for a sailing angle to the 10 -knot wind that produces maximum sailing speed. In this application, \((r, \theta)\) describes the sailing speed, \(r,\) in knots, at an angle \(\theta\) to the 10 -knot wind. Use this information to solve. Interpret the polar coordinates: \(\left(7.4,85^{\circ}\right)\)

Convert each polar equation to a rectangular equation. Then determine the graph's slope and y-intercept. $$r \sin \left(\theta-\frac{\pi}{4}\right)=2$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.