/*! 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 43 Solve the following equations. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 43

Solve the following equations. $$\cos 3 x=\sin 3 x, 0 \leq x<2 \pi$$

Short Answer

Expert verified
Answer: The specific solutions are \(x = \frac{\pi}{12}, \frac{3\pi}{12}, \frac{5\pi}{12}, ..., \frac{23\pi}{12}\).

Step by step solution

01

Use the identity \(\sin(y) = \cos(\frac{\pi}{2} - y)\)

Replace the sine function with its equivalent cosine function: $$\cos(3x) = \cos\left(\frac{\pi}{2} - 3x\right)$$
02

Apply the angle subtraction formula

When two cosine functions are equal, their angles either have to be equal directly or the sum of their angles equal to \(2n\pi\): $$3x = \frac{\pi}{2} - 3x \quad \text{or} \quad 3x + \frac{\pi}{2} - 3x = 2n\pi$$ For some integer \(n\).
03

Solve for x

Solve the equation for x and obtain the general solution: For the first case: $$3x = \frac{\pi}{2} - 3x$$ $$6x = \frac{\pi}{2}$$ $$x = \frac{\pi}{12}$$ For the second case: $$\frac{\pi}{2} = 2n\pi$$ $$n = \frac{1}{4}$$ However, n must be an integer, so the second case has no solution.
04

Find the specific solutions in the interval \([0, 2\pi)\)

Since the general solution is \(x = \frac{\pi}{12}\), the specific solutions will be multiples of this value within the interval: $$x = \frac{k\pi}{12},$$ where \(k = 0, 1, 2, ..., 24\). Observe that only the odd multiples of \(\frac{\pi}{12}\) will work, as only odd multiples will result in an angle subtraction of \(\frac{\pi}{2}\) which is necessary for the given equation. Therefore, the specific solutions are: $$x = \frac{\pi}{12}, \frac{3\pi}{12}, \frac{5\pi}{12}, ..., \frac{23\pi}{12}$$

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

Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. $$\sin ^{-1}(-1)$$

Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\sin ^{-1}(x / 3)\right)$$

The floor function, or greatest integer function, \(f(x)=\lfloor x\rfloor,\) gives the greatest integer less than or equal to \(x\) Graph the floor function, for \(-3 \leq x \leq 3\).

Sawtooth wave Graph the sawtooth wave defined by $$f(x)=\left\\{\begin{array}{ll} \vdots & \\\x+1 & \text { if }-1 \leq x<0 \\\x & \text { if } 0 \leq x<1 \\\x-1 & \text { if } 1 \leq x<2 \\\x-2 & \text { if } 2 \leq x<3 \\\\\vdots & \vdots\end{array}\right.$$

Volume of a spherical cap A single slice through a sphere of radius \(r\) produces a cap of the sphere. If the thickness of the cap is \(h,\) then its volume is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Graph the volume as a function of \(h\) for a sphere of radius \(1 .\) For what values of \(h\) does this function make sense?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.