/*! 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 17 Find all real numbers that satis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

Find all real numbers that satisfy each equation. $$ \sin x=\sqrt{2} / 2 $$

Short Answer

Expert verified
x = \( \frac{\pi}{4} + \2\pi k\ \) and \( \frac{3\pi}{4} + \2\pi k\ \), k \in \mathbb{Z}.

Step by step solution

01

Recall the value of \( \sin \theta =\sqrt{2} / 2 \

Note that \(\sin \theta \) equals \frac{\sqrt{2}}{2} \ when \theta \is \frac{\pi}{4} \ or \ \frac{3\pi}{4}\.
02

Write the general solutions

The sine function is periodic with a period of \(2\pi \). Therefore, the solutions in the general form are: \frac{\pi}{4} \+ \2\pi k \, k \in \mathbb{Z} \ and \frac{3\pi}{4} \+ \2\pi k \, k \in \mathbb{Z}\.
03

Combine the solutions

The complete set of solutions for the equation is \( \x = \frac{\pi}{4} + \2\pi k\ \) and \( \x = \frac{3\pi}{4} + \2\pi k\ \) where \ k \in \ \mathbb{Z}\.

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.

Sine Function
The sine function, often written as \(\sin x\)\, is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle. It ranges from -1 to 1 and is used to relate angles to side lengths in right triangles. Our equation is \(\sin x=\frac{\sqrt{2}}{2}\)\. To solve it, we look for angles where the sine value is \(\frac{\sqrt{2}}{2}\). This occurs commonly in 45-degree angles (\frac{\pi}{4} radians) and their equivalents.
General Solutions
When solving trigonometric equations like \(\sin x=\frac{\sqrt{2}}{2}\)\, it is important to understand the notion of 'general solutions'. The sine function is periodic, meaning it repeats its values in a regular pattern every \(2\pi\)\ radians. Thus, if \(\sin x = \frac{\sqrt{2}}{2}\)\ at \(\frac{\pi}{4}\)\ and \(\frac{3\pi}{4}\)\, all angles co-terminal with these solutions also solve the equation. We express this with \(\frac{\pi}{4} + 2\pi k\)\ and \(\frac{3\pi}{4} + 2\pi k\)\, where \(k\)\ is any integer.
Periodic Functions
Functions like sine that repeat values in a consistent interval are called periodic functions. For \(\sin x\)\, this period is \(2\pi\)\. This means no matter where you start, after moving along the x-axis by \(\2\pi\)\, the values of the function repeat. For example, \(\sin (x + 2\pi) = \sin x\)\. This property helps us find all possible solutions for trigonometric equations, as periodicity tells us that any original solution plus any multiple of \(2\pi\)\ is also a solution.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values in their domains and are crucial tools in solving trigonometric equations. To solve \(\sin x=\frac{\sqrt{2}}{2}\), we use the identity that notes sine's symmetry around the y-axis and its periodic nature. Recognizing that \(\frac{\sqrt{2}}{2}\) is a key value, we conclude \(\sin x = x\) is valid for \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\). Identifying these key angles is easier with knowledge of trigonometric identities and the unit circle.

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

Determine the amplitude, period, phase shift, and range for the function \(f(x)=3 \sin (x-\pi)+9\)

Solve each equation. Round approximate answers to the nearest tenth of a degree. $$ 3 \sin (\beta)+6=5 \sin (\beta)+7 \text { for } 0^{\circ} \leq \beta \leq 360^{\circ} $$

Solve each equation. Round approximate answers to the nearest tenth of a degree. $$ \frac{\sin \alpha}{23.4}=\frac{\sin 67.2^{\circ}}{25.9} \text { for } 0^{\circ}<\alpha<90^{\circ} $$

Find all real numbers that satisfy each equation. Round approximate answers to 2 decimal places. $$ \frac{\sin 49.6^{\circ}}{55.1}=\frac{\sin 88.2^{\circ}}{b} $$

Find the exact value of each composition without using a calculator or table. $$ \tan ^{-1}(\tan (\pi)) $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.