/*! 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 12 In Exercises 1-36, solve each of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 12

In Exercises 1-36, solve each of the trigonometric equations exactly on the interval \(0 \leq x<2 \pi\). $$ \sin (2 x)=\sqrt{3} \sin x $$

Short Answer

Expert verified
Solutions are \(x = 0, \pi, \frac{\pi}{6}, \frac{11\pi}{6}\).

Step by step solution

01

Use Double Angle Identity

Recall the double angle identity for sine: \( \sin(2x) = 2\sin(x)\cos(x) \). Substitute this into the given equation: \( 2\sin(x)\cos(x) = \sqrt{3}\sin(x) \).
02

Factor the Equation

Factor out \(\sin(x)\) from the equation: \( \sin(x)(2\cos(x) - \sqrt{3}) = 0 \). This equation is satisfied when either \(\sin(x) = 0\) or \(2\cos(x) - \sqrt{3} = 0\).
03

Solve for \(\sin(x) = 0\)

\(\sin(x) = 0\) at \(x = 0, \pi, 2\pi\) within the interval \(0 \leq x < 2\pi \). However, since the interval is \([0, 2\pi)\), the point \(2\pi\) is excluded. Thus, solutions are \(x = 0, \pi\).
04

Solve for \(2\cos(x) - \sqrt{3} = 0\)

Rearrange the equation: \(2\cos(x) = \sqrt{3}\). Thus, \(\cos(x) = \frac{\sqrt{3}}{2}\). \(\cos(x) = \frac{\sqrt{3}}{2}\) at \(x = \frac{\pi}{6}\) and \(x = \frac{11\pi}{6}\) within the interval \(0 \leq x < 2\pi\).
05

Compile All Solutions

Combining solutions from Step 3 and Step 4, the complete solution set is \(x = 0, \pi, \frac{\pi}{6}, \frac{11\pi}{6}\).

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.

Understanding the Double Angle Identity
The double angle identity for sine is a powerful tool used in trigonometry to simplify and solve equations. The identity is given by:\[ \sin(2x) = 2\sin(x)\cos(x) \]This equation expresses the sine of a double angle in terms of sine and cosine functions of the original angle. It is particularly useful because it breaks down more complicated expressions into simpler parts.
  • When you see an equation involving \(\sin(2x)\), consider substituting this identity.
  • This helps in transforming the equation into a form that can be more easily solved.
In our exercise, using the double angle identity allowed us to rewrite the equation \(\sin (2 x)=\sqrt{3} \sin x\) as \(2\sin(x)\cos(x) = \sqrt{3}\sin(x)\). This step was crucial in making the problem more approachable by breaking it down into a manageable form.
Demystifying the Sine Function
The sine function is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle. It's periodic with a period of \(2\pi\), which means it repeats every \(2\pi\) units. When solving trigonometric equations like \(\sin(x) = 0\), think about the points where the sine curve crosses the x-axis. These points are crucial as they indicate where the function equals zero.
  • \(\sin(x) = 0\) typically at multiples of \(\pi\): specifically at \(x = 0, \pi, 2\pi, \ldots\).
  • For an interval \(0 \leq x < 2\pi\), only include solutions at \(x = 0\) and \(x = \pi\), excluding \(2\pi\) because it is outside the specified interval.
Understanding these concepts helps solve trigonometric equations because it allows you to pinpoint exact values of x that satisfy the equation, improving your problem-solving efficiency.
Exploring the Cosine Function
Like sine, the cosine function is essential in trigonometry. It gives the x-coordinate of a point on the unit circle and is also periodic with a period of \(2\pi\).The cosine is particularly useful when solving equations where it's given as a specific value. For instance, finding \(x\) for which \(\cos(x) = \frac{\sqrt{3}}{2}\) involves remembering key angles where cosine takes on this value.
  • \(\cos(x) = \frac{\sqrt{3}}{2}\) at specific angles \(x = \frac{\pi}{6}\) and \(x = \frac{11\pi}{6}\) within \(0 \leq x < 2\pi\).
  • These angles correspond to the positions on the unit circle where the x-coordinate of the point is \(\frac{\sqrt{3}}{2}\).
Knowing these angles helps identify where certain trigonometric identities hold true, enabling you to solve equations more swiftly. By remembering these core values of the cosine function, one can solve trigonometric equations with greater ease and precision.

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

$$ \text { Solve }|\cos x| \geq \frac{1}{2} \text { over } 0 \leq x<2 \pi \text {. } $$

In Exercises 1-36, solve each of the trigonometric equations exactly on the interval \(0 \leq x<2 \pi\). $$ \frac{\cos (2 x)}{\cos x+\sin x}=0 $$

Optics. Assume that light is going from air into a diamond. Calculate the refractive angle \(\theta_{r}\) if the incidence angle is \(\theta_{i}=75^{\circ}\), and the index of refraction values for air and diamond are \(n_{i}=1.00\) and \(n_{r}=2.417\), respectively. Round to the nearest degree. (See Example 8 for Snell's law.)

In Exercises 1-36, solve each of the trigonometric equations exactly on the interval \(0 \leq x<2 \pi\). $$ \sqrt{3} \tan x=2 \sin x $$

Calculus. In calculus, the term "critical numbers" refers to values for \(x\), where a graph may have a maximum or minimum. To find the critical numbers of the equation \(y=\sin (2 x)+2\), solve the equation \(2 \cos (2 x)=0\) for all values of \(x\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.