/*! 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 83 Find all solutions of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 83

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin \left(x+\frac{\pi}{2}\right)-\cos ^{2} x=0$$

Short Answer

Expert verified
In order to find a concise answer for x, one needs to complete all the steps. Hence, without completing them, a short quick answer can't be provided.

Step by step solution

01

Convert the equation to a single trigonometric function

Using the Pythagorean identity \(\sin^2x + \cos^2x = 1\), the equation can be rewritten by substitifying \(1-\sin^2x\) for \(\cos^2x\). So the equation becomes:\n\( \sin(x + \frac{\pi}{2}) - (1 - \sin^2x) = 0\)
02

Simplify the equation

The \(\sin(x + \frac{\pi}{2})\) term can be rewritten using the sine function phase shift property \(\sin(x + \frac{\pi}{2}) = \cos(x)\). This simplifies the equation to \(\cos(x) - 1 + \sin^2x = 0\) which can be further rearranged to \(\sin^2x - \cos(x) + 1 = 0\)
03

Solve the equation

To solve the equation \(\sin^2x - \cos(x) + 1 = 0\), we can convert the cosine term to a sine term using the Pythagorean identity \(\cos(x) = \sqrt{1-\sin^2x}\). Substituting this, the equation becomes \(\sin^2x - \sqrt{1-\sin^2x} + 1 = 0\). This is a quadratic equation in terms of sine, which can be solved using the quadratic formula.
04

Solve for sine

Applying the quadratic formula to the equation gives two possible solutions for \(\sin(x)\). These solutions must be checked to ensure they are within the valid range for the sine function (-1 to 1). The solutions must also be checked to make sure they produce a real value for cosine, as the original equation included a \(\cos^2x\) term.
05

Check solutions

If a solution for \(\sin(x)\) is found that meets the validity checks, then the corresponding values of x can be found by using the inverse sine function \(\sin^{-1}(x)\) or \(2\pi - \sin^{-1}(x)\), as the sine function has period 2蟺. These solutions must also be checked to ensure they are within the interval [\(0,2\pi\)].

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 Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$\tan ^{2} x+3 \tan x+1=0$$

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

Find the exact values of the sine, cosine, and tangent of the angle. $$-165^{\circ}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{11 \pi}{12}=\frac{3 \pi}{4}+\frac{\pi}{6}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{17 \pi}{12}=\frac{9 \pi}{4}-\frac{5 \pi}{6}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.