By Dr. Shahid Ali Yousafzai

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As the Euler's formula is $${ e }^{ ix }=\cos { x } +i\sin { x }$$ Now if I replace $$x$$ here by $$i\pi$$ $${ e }^{ i\pi }=-1$$ $${ e }^{ i\pi }+1=0$$ This is the Euler's identity. It can be also written as $${ e }^{ i\pi }=\cos { \left( 1 \right) +i\sin { \left( 1 \right) } } =0.54+0.84i$$ Here the 1 is in radians, not degrees. We can plot the complex numbers as well but you have to keep in mind that in a unit circle we have $$r = 1$$, then $$x=\cos { \theta }$$ $$y=\sin { \theta }$$ By squaring we get $$\cos ^{ 2 }{ \theta } +\sin ^{ 2 }{ \theta } =1$$ Example: $$3+4i$$ $${ r }=\sqrt { { 3 }^{ 2 }+{ 4 }^{ 2 } } =5$$ $$\theta =\tan ^{ -1 }{ \frac { 4 }{ 3 } } =0.927$$

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