/*! 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 999 Find the equations for \(\sin 2 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 31: Problem 999

Find the equations for \(\sin 2 \theta\) and \(\cos 2 \theta\) from the de Moivre equation with \(\mathrm{n}=2\).

Short Answer

Expert verified
The equations for \(\sin(2\theta)\) and \(\cos(2\theta)\) using de Moivre's theorem with n=2 are: \(\sin(2\theta) = 2\cos\theta\sin\theta\) \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)

Step by step solution

01

Write down de Moivre's theorem for n=2

Since we want to use de Moivre's theorem with n=2, we can write the theorem as: \((\cos\theta + i\sin\theta)^2 = \cos(2\theta) + i\sin(2\theta)\)
02

Expand the left side of the equation

Now, let's expand the left side of the equation using the binomial theorem: \((\cos\theta + i\sin\theta)(\cos\theta + i\sin\theta) = \cos(2\theta) + i\sin(2\theta)\) Expanding the left side gives us: \(\cos^2\theta + 2i\cos\theta\sin\theta - \sin^2\theta = \cos(2\theta) + i\sin(2\theta)\)
03

Separate real and imaginary parts

Now we can separate the real and imaginary parts of the equation: Real part: \(\cos^2\theta - \sin^2\theta = \cos(2\theta)\) Imaginary part: \(2\cos\theta\sin\theta = \sin(2\theta)\)
04

Write the final equations

We have derived the equations for \(\sin(2\theta)\) and \(\cos(2\theta)\) using de Moivre's theorem. The final equations are: \(\sin(2\theta) = 2\cos\theta\sin\theta\) \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)

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

Find the four fourth roots of 9 . The polar form of 9 is 9 (cos \(\left.0^{\circ}+i \sin 0^{\circ}\right)\)

Find \([8(\cos \pi / 2+\mathrm{i} \sin \pi / 2)] \div[2(\cos \pi / 6+i \sin \pi / 6)]\)

A force of \(315 \mathrm{lbs}\). is acting at an angle of \(67^{\circ}\) with the horizontal. What are its horizontal and vertical components?

Show that \((-1 / 2+\sqrt{3 i / 2})^{3}=1\)

Use De moivre's theoram to find the value of \((\sqrt{3}+\mathrm{i})^{7}\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

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.