Why is eiθ=cos⁡(θ)+isin⁡(θ)?

Answered

2022-01-17

Why is eiθ=cos(θ)+isin(θ)?

Answer & Explanation

nick1337

nick1337

Expert

2022-01-17Added 573 answers

Step 1
Consider the function
f(x)=cosx+isinxeix
which converts a real x to an intricate number. Utilizing the quotient rule, determine the derivative with respect to x:
f(x)=(sinx+i×cosx)×eix(cosx+i×x)×i×eixeix2
Reordering the numerator proves: f(x)=0, in other words: f is a constant function; all reals are mapped to the same complex number. Which number? OK, try x=0; we know: cos0=1,sin0=0,e0=1; so f(0)=1, so f(x)=1 for all x, or
1=cosx+i×sinxeix
or
eix=cosx+i×sinx
QED.

star233

star233

Expert

2022-01-17Added 238 answers

Step 1
The reason this is true for most students nowadays is more so because their teachers told them this is true than anything else. But if you were asking to show that it is true using mathematics, then that is also possible to do.
We start by letting
z=cosθ+isinθ
z=sinθ+icosθ
Recall that i2=(1,0). That means we can replace the minus one with i2
z=i2sinθ+icosθ
Factoring out an i , we are left with:
z=i(cosθ+isinθ)
Recall that
cosθ+isinθ=zz=iz
How interesting. We see that this function must be such that its derivative is equal to itself multiplied by some constant. Doesn’t that sound oddly similar to the exponential function? Let’s keep going.
zz=i
We can now integrate both sides because we want to remove all of our derivatives.
zzdθ=idθ
The integral on the left is the world famous natural logarithm. The integral on the right is trivial because we are just integrating some constant function.
lnz=iθ+C
Recall now the definition of the logarithm.
z=eiθ+C
Wow, looks like we’re starting to get there!
z=eiθeC
I rewrote eC as just C because they’re both just some constants of integration.
z=C×eiθ
We’re so close, but we have to find the value for the constant of integration before we go anywhere else. We’ll start by replacing z for what it is substituting.
cosθ+isinθ=C×eiθ
Let's let θ=0. In doing so, we see that a lot of things cancel nicely.
cos0+isin0=C×e0
1=C
Now we know the value for the constant, we can go back to the equation we had earlier.
z=C×eiθ
Now, to make all of our proper substitutions.
cosθ+isinθ=eiθ
And now we know why this is true.

alenahelenash

alenahelenash

Expert

2022-01-24Added 366 answers

Step 1 The most common way is to calculate the Taylor series for each: exp(z)=1+z11!+z22!+z33!+z44!+z55!+ cosz=1z22!+z44!z66!+z88!z1010!+ sinz=z11!=z33!+z55!z77!+z99!z1111!+ Now calculate the series for exp(iz) and see how it compares when you add the series for cosz and isinz .

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