/*! 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 34 In Exercises 1-36, solve each of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 34

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

Short Answer

Expert verified
The solution is \( x = \frac{\pi}{2} \).

Step by step solution

01

Apply Trigonometric Identity

Recognize that the equation \( \sin (3x) \cos (2x) - \cos (3x) \sin (2x) = 1 \) matches the form of the sine difference identity: \( \sin(a-b) = \sin a \cos b - \cos a \sin b \). Therefore, we can rewrite the left-hand side as \( \sin((3x) - (2x)) \).
02

Simplify the Expression

Substitute the simplified trigonometric identity into the equation: \( \sin(x) = 1 \). The original equation simplifies to \( \sin(x) = 1 \).
03

Find Values of x

Determine the values of \( x \) that satisfy \( \sin(x) = 1 \) within the interval \([0, 2\pi)\). The sine function equals 1 at \( x = \frac{\pi}{2} \).
04

Conclude the Solution

Since \( x = \frac{\pi}{2} \) is the only solution within \([0, 2\pi)\), we conclude this is the only satisfying \( x \).

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.

Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are powerful tools in simplifying complex trigonometric expressions. In the given problem, we utilize the sine difference identity, which is written as \( \sin(a-b) = \sin a \cos b - \cos a \sin b \). This identity allows us to rewrite the expression \( \sin(3x) \cos(2x) - \cos(3x) \sin(2x) \) as \( \sin((3x) - (2x)) \).
This transformation simplifies the original equation significantly:
  • It reduces the left-hand side to a single sine function of a single angle, \( \sin(x) \).
  • By using this identity, we make the equation easier to tackle, setting the stage for straightforward solution steps.
Understanding and recognizing these identities can greatly simplify solving trigonometric equations, as demonstrated in this exercise.
Sine Function
The sine function, denoted as \( \sin \), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. On the unit circle, it represents the y-coordinate of a point corresponding to an angle \( x \) measured from the positive x-axis.
For the problem at hand, we focus on the equation \( \sin(x) = 1 \). The sine function reaches its maximum value of 1 at a specific point:
  • Sine of any angle \( x \) within the range \( 0 \leq x < 2\pi \) is 1 only at \( x = \frac{\pi}{2} \).
  • This value corresponds to a point at the peak of the sine wave, where the y-coordinate is precisely at the highest point.
Understanding where the sine function achieves its peak is crucial for determining the correct values of \( x \) that satisfy the given equation.
Interval Notation
Interval notation is a mathematical shorthand used to denote a range of values between two endpoints. It is useful in expressing solutions or domains of functions, indicating which values are included or excluded.
In our case, we are solving for \( x \) in the interval \([0, 2\pi)\). This notation tells us:
  • \([0, 2\pi)\) includes all real numbers \( x \) starting from 0 up to but not including \( 2\pi \).
  • The bracket \([\) means that 0 is included in the interval, while the parenthesis \()\) signifies that \( 2\pi \) is not included.
By understanding interval notation, we ensure that our solution, \( x = \frac{\pi}{2} \), falls squarely within the specified boundaries and adheres to the problem's constraints.

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

Solve for the smallest positive \(x\) that makes this statement true: $$ \sin \left(x+\frac{\pi}{4}\right)+\sin \left(x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2} $$

For Exercises 57 and 58 , refer to the following: Allergy sufferers' symptoms fluctuate with the concentration of pollen in the air. At one location the pollen concentration, measured in grains per cubic meter, of grasses fluctuates throughout the day according to the function: $$ p(t)=35-26 \cos \left(\frac{\pi}{12} t-\frac{7 \pi}{6}\right), \quad 0 \leq t \leq 24 $$ where \(t\) is measured in hours and \(t=0\) is \(12: 00\) A.M. Biology/Health. Find the time(s) of day when the grass pollen level is 41 grains per cubic meter. Round to the nearest hour.

Optics. Assume that light is going from a diamond into air. Calculate the refractive angle \(\theta_{r}\) if the incidence angle is \(\theta_{i}=15^{\circ}\), and the index of refraction values for diamond and air are \(n_{i}=2.417\) and \(n_{r}=1.00\), respectively. Round to the nearest degree. (See Example 8 for Snell's law.)

For Exercises 87-92, refer to the following: Graphing calculators can be used to find approximate solutions to trigonometric equations. For the equation \(f(x)=g(x)\), let \(Y_{1}=f(x)\) and \(Y_{2}=g(x)\). The \(x\) values that correspond to points of intersections represent solutions. Use a graphing utility to find all solutions to the equation \(\cos \theta=e^{\theta}\) for \(\theta \geq 0\).

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

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.