/*! 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 93 Use words (not an equation) to d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 93

Use words (not an equation) to describe one of the Pythagorean identities.

Short Answer

Expert verified
The Pythagorean identity involving sine and cosine states that for any given angle in a right triangle, the sum of the square of the sine of the angle and the square of the cosine of the same angle always equals one.

Step by step solution

01

Identify a Pythagorean Identity

To start, we'll choose one of the Pythagorean identities to describe. Let's select the fundamental identity: the square of the sine of an angle plus the square of the cosine of the same angle equals one.
02

Break down the equation components

This identity includes: sine, cosine, angle, square and the constant one. Each component needs to be explained separately - sine and cosine as ratios defining the relationship between sides in a right triangle for a given angle, and the concept of squaring as multiplying a number by itself.
03

Describe the Pythagorean Identity

The chosen Pythagorean identity can be verbally explained as follows: if we take an angle in a right triangle and square the length ratio of the opposite side to the hypotenuse (which is the sine of the angle), then add it to the square of the ratio of the adjacent side to the hypotenuse (which is the cosine of the angle), the result will always be one. This holds true regardless of the size of the angle, as long as it is in a right triangle.

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: Understanding the Basics
Trigonometry is a branch of mathematics concerned with the relationships between the angles and sides of triangles. At its core, it helps us understand the properties of triangles, especially right triangles. This branch connects angles to the ratios of different sides of a triangle, introducing us to functions like sine, cosine, and tangent.
  • The right triangle is central to trigonometry; by using trigonometric functions, we can solve for various lengths and angles within it.
  • Each angle's measure determines the sine (ratio of opposite side to hypotenuse) and cosine (ratio of adjacent side to hypotenuse).

Trigonometry is used in a broad range of applications, from engineering and physics to computer graphics. By using these trigonometric ratios, complex problems in various fields can be simplified and worked out effectively.
Right Triangle: The Foundation of Trigonometry
A right triangle is a special type of triangle where one angle always measures exactly 90 degrees. The sides of the right triangle hold notable importance in trigonometry.
  • There are three sides in a right triangle: the hypotenuse (the longest side opposite the right angle), the opposite side (opposite to the specified angle), and the adjacent side (next to the specified angle).
  • Understanding these sides helps us apply trigonometric ratios like sine and cosine efficiently.

In right triangles, the relationships between these sides and angles provide a logical basis for various trigonometric functions and identities, such as the Pythagorean identities.
Sine and Cosine: Key Trigonometric Functions
Sine and cosine are fundamental trigonometric functions that help describe the relationship between the angles and sides of right triangles.
  • The sine of an angle (\( heta \)) in a right triangle is calculated as the length of the opposite side divided by the hypotenuse: \( ext{sin}( heta) =\, \frac{\text{opposite side}}{\text{hypotenuse}} \).
  • The cosine of the same angle is the length of the adjacent side divided by the hypotenuse: \( ext{cos}( heta) =\, \frac{\text{adjacent side}}{\text{hypotenuse}} \).

These ratios help solve different problems involving right triangles, allowing us to find unknown side lengths and angles. Furthermore, they play a crucial role in defining the Pythagorean identity, which states that the square of the sine of an angle plus the square of the cosine of the same angle equals one (\( ext{sin}^2(\theta) + ext{cos}^2(\theta) = 1 \))鈥攁 key concept for understanding how these functions are interrelated.

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

Make Sense? In Exercises \(116-119\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \(y=\sin x\) has an inverse function if \(x\) is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) they should make restrictions easier to remember by also using \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) as the restriction for \(y=\tan x\)

Let \(f(x)=\left\\{\begin{array}{ll}x^{2}+2 x-1 & \text { if } x \geq 2 \\ 3 x+1 & \text { if } x<2\end{array}\right.\) Find \(f(5)-f(-5) .\) (Section 1.3, Example 6)

Graph: \(x^{2}+y^{2}=1 .\) Then locate the point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) on the graph.

Solve \(y=2 \sin ^{-1}(x-5)\) for \(x\) in terms of \(y\)

What do we mean by even trigonometric functions? Which of the six functions fall into this category?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.