By Dr. Shahid Ali Yousafzai

6 Views

Let’s discuss the Euler's formula which is $$ { e }^{ ix }=\cos { x } +i\sin { x } $$ It can also be written as $$ { e }^{ i\theta }=\cos { \theta } +i\sin { \theta } $$ This \(i\) is defined as $$ { i }^{ 2 }=-1 $$ It is an imaginary and unique number and it is unique because $$ 1\times i=-1 $$ No such other identical number exists such that we multiply it and we get -1. We know this series $$ { e }^{ x }=1+\frac { x }{ 1! } +\frac { { x }^{ 2 } }{ 2! } +\frac { { x }^{ 3 } }{ 3! } +...+\frac { { x }^{ n } }{ n! } $$ which is expanded by Taylor's series. Now if I replace this \(x\) with \(ix\), we get $$ { e }^{ ix }=1+\frac { ix }{ 1! } +\frac { { ix }^{ 2 } }{ 2! } +\frac { { ix }^{ 3 } }{ 3! } +\frac { { ix }^{ 4 } }{ 4! } +... $$ $$ { e }^{ ix }=1+ix-\frac { { x }^{ 2 } }{ 2! } -\frac { { ix }^{ 3 } }{ 3! } +\frac { { x }^{ 4 } }{ 4! } +... $$ If we split it now $$ { e }^{ ix }=1-\frac { { x }^{ 2 } }{ 2! } +\frac { { x }^{ 4 } }{ 4! } +...+ix-i\left( \frac { { x }^{ 3 } }{ 3! } +... \right) $$ This gives us the expansion of \(\cos { x } \) and \(\sin { x } \). Thus $$ { e }^{ ix }=\cos { x } +i\sin { x } $$ Or $$ { e }^{ i\theta }=\cos { \theta } +i\sin { \theta } $$ This is called Euler's formula. Now if I consider a circle in cartesian coordinates, then $$ \frac { x }{ r } =\cos { \theta } $$ $$ \frac { y }{ r } =\sin { \theta } $$ $$ x=r\cos { \theta } $$ $$ y=r\sin { \theta } $$ Squaring them $$ { x }^{ 2 }+{ y }^{ 2 }={ r }^{ 2 }\cos ^{ 2 }{ \theta } +{ r }^{ 2 }\sin ^{ 2 }{ \theta } $$ Taking \(r^2\) common $$ { r }^{ 2 }\left( \cos ^{ 2 }{ \theta } +\sin ^{ 2 }{ \theta } \right) ={ x }^{ 2 }+{ y }^{ 2 } $$ $$ r=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } $$ Or, it can also be done as $$ \frac { y }{ x } =\frac { r\sin { \theta } }{ r\cos { \theta } } =\tan { \theta } $$ $$ \theta =\tan ^{ -1 }{ \frac { y }{ x } } $$ Cartesian coordinates \((x,y)\) and polar coordinates \((r, \theta )\) are interconvertible.

Latest News

  • Become an Instructor 4 March, 2018

    Apply to join the passionate instructors who share their expertise and knowledge with the world. You'll collaborate with some of the industry's best producers, directors, and editors so that your content is presented in the best possible light..

More Chapters

  • Introduction to Physics

    • Other Subjects

  • English
  • Applied Physics
  • Environmental Studies
  • Physical Chemistry
  • Analytical Chemistry
  • Organic Chemistry
  • Soft Skills
  • Engineering Drawing
  • General Medicine
  • Mathematics
  • Patente B Italia
  • Adult Education
  • Engineering Chemistry
  • Conceptual Physics