/*! 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 50 Involve trigonometric equations ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 50

Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$ \sec ^{2} x-2=0 $$

Short Answer

Expert verified
The solutions for the original equation \(\sec^2 x - 2 = 0\) within the interval [0, 2蟺) are \(x = \frac{\pi}{4}, \frac{7\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\).

Step by step solution

01

Rewrite secant as reciprocal of cosine

Rewrite the secant (\(\sec\)) function in the equation as the reciprocal of cosine (\(\cos\)). This gives: \(\frac{1}{\cos^2x} - 2 = 0\).
02

Solve for \(\cos^2x\)

Isolate \(\cos^2x\) on one side of the equation: \(\cos^2x = \frac{1}{2}\).
03

Take square root on both sides

Since \(\cos^2x\) is equal to \(\frac{1}{2}\), taking the square root on both sides gives us \(\cos x = 卤\sqrt{\frac{1}{2}} = 卤\frac{1}{\sqrt{2}} = 卤\frac{\sqrt{2}}{2}\).
04

Find all solutions on the interval [0, 2蟺)

The solutions in the interval [0, 2蟺) for \(\cos x = \frac{\sqrt{2}}{2}\) are \(x = \frac{\pi}{4}, \frac{7\pi}{4}\) and for \(\cos x = -\frac{\sqrt{2}}{2}\) are \(x = \frac{3\pi}{4}, \frac{5\pi}{4}\).
05

Compilation of solutions

So, all the solutions for the original equation \(\sec^2 x - 2 = 0\) within the interval [0, 2蟺) are \(x = \frac{\pi}{4}, \frac{7\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\).

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

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \cos ^{2} x+2 \cos x-2=0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I solved \(4 \cos ^{2} x=5-4 \sin x\) by working independently with the left side, applying a Pythagorean identity, and transforming the left side into \(5-4 \sin x .\)

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \cos x-5=3 \cos x+6 $$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \cos x \csc x=2 \cos x $$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \cos ^{2} x+5 \cos x-1=0 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.