Eulers formula

Euler's formula relates the complex exponential to the cosine and sine functions. This formula is the most important tool in AC analysis. It is why electrical engineers need to understand complex numbers.

[imrozzahan]

This notation and formula is used in quantum mechanics and wave function. Amazingly beautiful. The mathematics euler invented three centuries ago, is now used in an elegant physics theory like quantum mechanics.

[rafiqulislam]

Prove that ( in Euler's formula),
sin^2(x) + cos^2(x) = 1
Solution:-
We know that,
e^(ix) = cosx +
e^(-ix) = cosx -
[ 'i' is the complex number]
Then multiply both equations,
e^(ix-ix) = (cosx + isinx)(cosx - isinx)
or, e^0 = (cosx)^2 - (isinx)^2
or, 1 = (cosx)^2 - (i^2)(sinx)^2
[As, (i^2) = -1]
So, 1= (cosx)^2 - (-1)(sinx)^2
or, 1= (cosx)^2 + (sinx)^2
Thus, sin^2(x) + cos^2(x) = 1
😁😁😁

[Kevin-rvbr]

Why do electrical engineers use j instead of i as the immaginary unit?

[GammaDigamma]

And if we let j=1+0i as 1 is an complex number, we get the definitions of hyperbolic trigonometric functions, it all finally makes sense

[EASTSIDESEE]

Wowsers, I just heard terms that made so much more sense (just having been exposed to Calc over the past 9 years). Amazing explanations as I have been struggling to fully grasp the why of this formula. Thank you.

[spaceengineering]

Dear Khan Academy,

Can you please put a link to the video giving the proof of the euler equation which you referenced in this video.

Thank you

Appreciate it.

[mainethang]

After seeing this, I’m now chasing my football career. What is all this.

[alejrandom]

I'm a simple man, I hear "we love e", I hit like

[omarrasouli]

I actually understood this. ayy I'm going to be doing this in 3 months time...

[JesusMartinez-zuxl]

thank you! Eulers formula is mentioned within my matlab class so i just wanted to learn a bit more. Thank you once again!

[bicresty]

Does anyone have a link to the video with the proof for Euler's formula?

[arunsinghmehra]

The way you teach is damn
Yeah I love e

[curtpiazza]

WOW! Great quick video...very well presented and explained! Videos of this quality help me fill in the gaps in my math background. ❤ 😊

[srivatsaa.r.]

KHAN ACADEMY
E U L E R S F O R M U L E A

Just watched yr above video. Liked the video and learnt EULERS FORMULA and associated formulas to keep in mind.
Clearly audible and clearly visible even in my small mobile screen.
CLEAR with out a doubt.
Thanks
Vatsa BANGALORE INDIA

[putinscat]

I prefer the taylor's series proof of this. Then it makes more sense, instead of wondering "how the F does a cosine = an e ? "

[aysenur]

Sometimes I feel like professors are just gatekeepers... Thank you for the video!

[jianlin]

When i heard Euler I knew I shouldn’t be here, Still in Algebra 2!! Don’t mind me!

[nlanyweb]

where is the Sal's video that proves Euler's formula?

[shivanisingh]

Before seeing this video I love X but now I love E ❤️😅

[andrewdeleonardis]

Is this essentially putting trig functions in terms of complex hyperbolic trig functions?