Question

solve equation: sinx+icosx=cosx+isinx

Answer 1

Well all the imaginary parts must be equal to the imaginary parts since you can't get an imaginary number from sine or cosine.

sinx=cosx
icosx=isinx

So after splitting it up into two parts you can see that dividing the second one by i on both sides just gets you really only one equation

sinx=cosx

And when is that true? First at 45 degrees then 180 degrees away from that. And of course it's periodic so it happens at all these points.

Answer 2

oh,i know it,but i dont think that it is the best way(

Answer 3

x=45 degree

Answer 4

thank you very much for helping

Answer 5

An alternative way is if you know \[e^{ix}=cosx+isinx\]
then you can see that \[ie^{-ix}=e^{ix}\]
then you can see that negative exponents are in the denominator so:\[\frac{ i }{ e^{ix} }=e^{ix}\]so then you have:\[\sqrt{i} = e^{ix}\]notice that i and square roots can also be represented as an exponential \[e^{i \frac{ \pi }{ 2 }*\frac{ 1 }{ 2 }}=e^{ix}\] Since the logarithm of a complex number is cyclic by 2pi, you can see that \[n2\pi+\frac{ \pi }{ 4 }=x\]
But wait there are two answers! That's alright, just remember when you take the square root above you really get a positive and negative answer.

Answer 6

seems like Kainui is correct, there r 2 answers !

Answer 7

Yeah you can have sin(x)=cos(x) when they are:

\[\frac{ \sqrt{2} }{ 2 }=\frac{ \sqrt{2} }{ 2 }\] and \[-\frac{ \sqrt{2} }{ 2 }=-\frac{ \sqrt{2} }{ 2 }\]

Answer 8

so ans = n*1/underoot 2

Answer 9

No this is when the sine and cosine functions are equal. The angle at which they are these is when

Answer 10

ohh

Answer 11

thanks)) ,but i have one more problem( can i ask you?