/*! 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 2 State the three Pythagorean iden... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 2

State the three Pythagorean identities.

Short Answer

Expert verified
Answer: The three Pythagorean identities are: 1. \((\sin{\theta})^2 + (\cos{\theta})^2 = 1\), derived from the Pythagorean theorem applied to a right triangle with a hypotenuse of length 1. 2. \((\tan{\theta})^2 + 1 = (\sec{\theta})^2\), derived by dividing both sides of the first identity by \((\cos{\theta})^2\). 3. \(1 + (\cot{\theta})^2 = (\csc{\theta})^2\), derived by dividing both sides of the first identity by \((\sin{\theta})^2\). These identities are trigonometric relationships based on the Pythagorean theorem and are true for any angle in a right triangle.

Step by step solution

01

Pythagorean Identity 1 (Sine & Cosine)

The first Pythagorean identity is derived from the Pythagorean theorem applied to a right triangle with a hypotenuse of length 1. In this case, the theorem states that the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse, i.e., \((\sin{\theta})^2 + (\cos{\theta})^2 = 1\). This identity is true for any angle \(\theta\).
02

Pythagorean Identity 2 (Tangent & Secant)

The second Pythagorean identity is derived from the first identity. By dividing both sides of the first identity by \((\cos{\theta})^2\), we get the following identity: \[\frac{(\sin{\theta})^2}{(\cos{\theta})^2} + \frac{(\cos{\theta})^2}{(\cos{\theta})^2} = \frac{1}{(\cos{\theta})^2}.\] Since \(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\) and \(\sec{\theta} = \frac{1}{\cos{\theta}}\), we can rewrite the above equation as \((\tan{\theta})^2 + 1 = (\sec{\theta})^2\). This identity is also true for any angle \(\theta\).
03

Pythagorean Identity 3 (Cotangent & Cosecant)

The third Pythagorean identity is derived by dividing both sides of the first identity by \((\sin{\theta})^2\). In this case, we get the following identity: \[\frac{(\sin{\theta})^2}{(\sin{\theta})^2} + \frac{(\cos{\theta})^2}{(\sin{\theta})^2} = \frac{1}{(\sin{\theta})^2}.\] Since \(\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}}\) and \(\csc{\theta} = \frac{1}{\sin{\theta}}\), we can rewrite the above equation, obtaining \(1 + (\cot{\theta})^2 = (\csc{\theta})^2\). This identity is true for any angle \(\theta\). So the three Pythagorean identities are: 1. \((\sin{\theta})^2 + (\cos{\theta})^2 = 1\) 2. \((\tan{\theta})^2 + 1 = (\sec{\theta})^2\) 3. \(1 + (\cot{\theta})^2 = (\csc{\theta})^2\)

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

Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(7 .\) Make \(a\) table similar to Table 7.8 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{0}^{4}\left(3 x^{5}-8 x^{3}\right) d x=1536\)

A differential equation and its direction field are given. Sketch a graph of the solution that results with each initial condition. $$\begin{aligned}&y^{\prime}(t)=\frac{\sin t}{y},\\\&y(-2)=-2 \text { and }\\\&y(-2)=2\end{aligned}$$

Approximate the following integrals using Simpson's Rule. Experiment with values of \(n\) to ensure that the error is less than \(10^{-3}\). \(\int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x=\frac{2 \pi}{3}\)

Arc length of a parabola Let \(L(c)\) be the length of the parabola \(f(x)=x^{2}\) from \(x=0\) to \(x=c,\) where \(c \geq 0\) is a constant. a. Find an expression for \(L\) and graph the function. b. Is \(L\) concave up or concave down on \([0, \infty) ?\) c. Show that as \(c\) becomes large and positive, the arc length function increases as \(c^{2} ;\) that is, \(L(c) \approx k c^{2},\) where \(k\) is a constant.

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. $$y^{\prime}(t)=\sin y, y(-2)=\frac{1}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.