Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
4th Edition
ISBN: 9781337687805
Author: James Stewart
Publisher: Cengage Learning
bartleby

Concept explainers

bartleby

Videos

Textbook Question
Chapter I, Problem 48E

Use Euler’s formula to prove the following formulas for cos x and sin x:

cos x = e i x + e i x 2 sin x = e i x e i x 2 i

Expert Solution & Answer
Check Mark
To determine

To prove The formulas cosx=eix+eix2 and sinx=eixeix2i by the use of Euler formula.

Explanation of Solution

Formula used:

Euler’s formula:

eiy=cosy+isiny

Calculation:

Simplify the expression eix+eix by the use of Euler’s formula as follows.

eix+eix=(cosx+isinx)+(cos(x)+isin(x))=cosx+isinx+cosxisinx[cos(x)=cosxandsin(x)=sinx]=cosx+cosx=2cosx

Thus, cosx=eix+eix2.

Similarly, simplify the expression eixeix by the use of Euler’s formula as follows.

eixeix=(cosx+isinx)(cos(x)+isin(x))=cosx+isinxcosx+isinx[cos(x)=cosxandsin(x)=sinx]=isinx+isinx=2isinx

Thus, sinx=eixeix2i.

Additional Math Textbook Solutions

Find more solutions based on key concepts
Knowledge Booster
Calculus
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
  • Calculus: Early Transcendentals
    Calculus
    ISBN:9781285741550
    Author:James Stewart
    Publisher:Cengage Learning
    Thomas' Calculus (14th Edition)
    Calculus
    ISBN:9780134438986
    Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
    Publisher:PEARSON
    Calculus: Early Transcendentals (3rd Edition)
    Calculus
    ISBN:9780134763644
    Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
    Publisher:PEARSON
  • Calculus: Early Transcendentals
    Calculus
    ISBN:9781319050740
    Author:Jon Rogawski, Colin Adams, Robert Franzosa
    Publisher:W. H. Freeman
    Precalculus
    Calculus
    ISBN:9780135189405
    Author:Michael Sullivan
    Publisher:PEARSON
    Calculus: Early Transcendental Functions
    Calculus
    ISBN:9781337552516
    Author:Ron Larson, Bruce H. Edwards
    Publisher:Cengage Learning
  • Calculus: Early Transcendentals
    Calculus
    ISBN:9781285741550
    Author:James Stewart
    Publisher:Cengage Learning
    Thomas' Calculus (14th Edition)
    Calculus
    ISBN:9780134438986
    Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
    Publisher:PEARSON
    Calculus: Early Transcendentals (3rd Edition)
    Calculus
    ISBN:9780134763644
    Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
    Publisher:PEARSON
    Calculus: Early Transcendentals
    Calculus
    ISBN:9781319050740
    Author:Jon Rogawski, Colin Adams, Robert Franzosa
    Publisher:W. H. Freeman
    Precalculus
    Calculus
    ISBN:9780135189405
    Author:Michael Sullivan
    Publisher:PEARSON
    Calculus: Early Transcendental Functions
    Calculus
    ISBN:9781337552516
    Author:Ron Larson, Bruce H. Edwards
    Publisher:Cengage Learning
    Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY