/*! 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 3 Find all solutions of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 3

Find all solutions of the equation. $$2 \sin x-1=0$$

Short Answer

Expert verified
\( x = \frac{\pi}{6} + 2k\pi \) or \( x = \frac{5\pi}{6} + 2k\pi \), for any integer \( k \).

Step by step solution

01

Isolate the sine function

Start with the equation \( 2 \sin x - 1 = 0 \). Add 1 to both sides to get \( 2 \sin x = 1 \). Then, divide both sides by 2 to isolate the sine function: \( \sin x = \frac{1}{2} \).
02

Find the general solution for \( \sin x = \frac{1}{2} \)

Recall that \( \sin x = \frac{1}{2} \) at angles \( x \) where \( x = \frac{\pi}{6} + 2k\pi \) or \( x = \frac{5\pi}{6} + 2k\pi \), where \( k \) is any integer. This is because \( \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \) and \( \sin \left(\frac{5\pi}{6}\right) = \frac{1}{2} \).
03

Write the solution set

Combine the results from Step 2. The solutions are given by \( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{5\pi}{6} + 2k\pi \), for any integer \( k \). Thus, the solution set consists of all angles that are equivalent to these expressions.

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 denoted as \( \sin(x) \), is one of the fundamental components of trigonometry. It's a periodic function that occurs naturally in various contexts, such as wave functions in physics. The sine of an angle in a right triangle is defined as the opposite side over the hypotenuse. But when we extend this definition to all angles, we use a unit circle, where:
  • The circle has a radius of 1.
  • The angle \( x \) is measured from the positive x-axis.
  • The sine of the angle is the y-coordinate of the corresponding point on the circle.
The sine function varies between -1 and 1 and has a period of \( 2\pi \) radians, meaning every \( 2\pi \) radians, the sine wave repeats itself.
General Solution
A general solution in trigonometry contains all possible solutions to an equation. When dealing with sinusoidal functions like \( \sin(x) \), we utilize the periodic nature to express solutions.For instance, solving \( \sin x = \frac{1}{2} \) involves:
  • Identifying specific solutions on the unit circle where the sine of these angles equals \( \frac{1}{2} \).
  • For \( \sin x = \frac{1}{2} \), the angles are \( \frac{\pi}{6} \) and \( \frac{5\pi}{6} \) radians, noted from the unit circle.
  • Because the sine function repeats every \( 2\pi \), the general solutions are expressed as \( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{5\pi}{6} + 2k\pi \), where \( k \) is an integer.
      • These solutions cover all possible angles which satisfy the equation given the periodic nature of sine.
Radians
Radians are a way to measure angles based on the radius of a circle. One full rotation around a circle corresponds to \( 2\pi \) radians, which is equal to 360 degrees. Here's how radians work:
  • A straight angle (or half a circle) is \( \pi \) radians, equivalent to 180 degrees.
  • Thus, \( \frac{\pi}{6} \) radians is 30 degrees, and \( \frac{5\pi}{6} \) radians is 150 degrees.
Using radians simplifies many calculations in calculus and trigonometry because they relate directly to the geometry of circles. For instance, when solving trigonometric equations, expressing angles in radians allows for more straightforward application of formulas and exploration of their periodic properties.

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

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7 ). Assume \(0 \leq \theta<\pi / 2.\) $$\frac{\sqrt{x^{2}-25}}{x}, \quad x=5 \sec \theta$$

Use a graphing device to find the solutions of the equation, correct to two decimal places. $$\frac{\cos x}{1+x^{2}}=x^{2}$$

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=\cos ^{4} x-\sin ^{4} x, \quad g(x)=2 \cos ^{2} x-1$$

Use a double- or half-angle formula to solve the equation in the interval \([0,2 \pi)\). $$\tan \frac{x}{2}-\sin x=0$$

Find the exact value of the expression, if it is defined. $$\tan \left(\sin ^{-1} \frac{1}{2}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

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.