/*! 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 77 Explain why the equation is not ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 77

Explain why the equation is not an identity and find one value of the variable for which the equation is not true. $$1+\tan \theta=\sec \theta$$

Short Answer

Expert verified
The equation is not an identity as it's not true for all values of \(\theta\). For example, this equation does not hold true for \(\theta = \frac{\pi}{2}\).

Step by step solution

01

Understand the Trigonometric Identities

The equation given is \(1+\tan\theta=\sec\theta\). In order to test if this is an identity, let's first understand what the trigonometric functions tangent (\(\tan\)) and secant (\(\sec\)) mean. The \(\tan\theta\) function is equivalent to \(\frac{\sin\theta}{\cos\theta}\) and the \(\sec\theta\) function is equivalent to \(\frac{1}{\cos\theta}\).
02

Substitute identities into the equation

We substitute these identities into the initial equation, obtaining: \(1+\frac{\sin\theta}{\cos\theta}=\frac{1}{\cos\theta}\). We simplify this to: \(\frac{1+\sin\theta}{\cos\theta}=\frac{1}{\cos\theta}\)
03

Choose a value for theta that invalidates the equation

We can see that for the equation to hold, \(1+\sin\theta\) must always equal to 1. One of the easy choices for \(\theta\) is any value where \(\sin\theta \neq 0\). Let's choose \(\theta = \frac{\pi}{2}\). Substituting in the equation, we find: \(1+ \sin\frac{\pi}{2}\neq \sec\frac{\pi}{2}\), because \(\sin\frac{\pi}{2} =1 \) and \(\sec\frac{\pi}{2} = \) is undefined. Therefore, the initial equation is not an identity.

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 the \(x\) -intercepts of the graph. $$y=\sin \frac{\pi x}{2}+1$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$x \cos x-1=0$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$4 \sin ^{3} x+2 \sin ^{2} x-2 \sin x-1=0$$

Solve the multiple-angle equation. $$\sec 4 x=2$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos x+\sin x \tan x=2$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.