Even if you have no idea what on earth any of this stuff is but have a vast knowledge in complex number system you might be able to help me as this is what I believe is preventing me from making sense of this stuff...
Ok I'm trying to understand how you would know that sin(pi/4) is equal to sqrt(2)/2 if you were never told that it was so... not doing any gemoetric proofs... yes I know there are some for those angles... instaed I would like to turn to works of euler and try to understand that way...
According to Euler... yes this question has an answer as the formula has been around for a couple hundred of years and somebody should be help me understand...
sin(x) = ( e^(ix) - e^(-ix) )/(2i) or solemly noted ever as ( cis(x) - cis(-x) )/(2i) which one you prefer
I was woundering if somebody as well could tell me what cis stands for
like you know how sin(x) is spoken sine of x what is cis
I thought it might be like coimaginarysine or something
I have always been told that cis(x)= e^(ix) = meh don't feel like fliping thorugh my notebook but ya you should know what that is also equal to and was never told how to pronounce cis
fine
e^(ix) = cos(x) = cosx + i sin(x)
e^(-ix) = cos(x) - i sin(x) = 1/e^(ix)
anways inspection formula yield me to come to the conclusions that sense in the formlua
( e^(ix) - e^(-ix) )/(2i)
were dividing by 2 here the numerator should yield the lenght of the unit circle that the sine of the angle refers to times 2 times i
sure enough pluging into your calculator the numerator the formula with x being pi/4 you get some number and putting in isqrt(2) you get exactly the same just as I had sepeculated, twice the lenght on the unit circle coresponding to that angle times the imaginary number because sin is a reference to the second dimension, the set of complex numbers... hence dividng that number by 2i you get none other then sqrt(2)/2 which makes perfect sense...
now my question is how would I know what the numerator equal isqrt(2) if I didn't have my calculator how do I simplify the numerator pluging in pi/4 for x and getting isqrt(2) I was woundering if you could show me how to do this...
this may help refresh your memory on some things
e^(ipi) = -1
ipi = ln(-1)
pi = ln(-1)/i
i = ln(-1)/pi = sqrt(-1)
as I stated early this stuff has been around for several centuries especially the complex number system which my lack of knowledge is hindering my ability to simplify the numerator which I am hopeing somebody on here could help me with
Also one other thing that spooked me was if pluging pi/4 into this equation
e^(ix) - e^(-ix) = isqrt(2)
and I divide through by i I have the exact value of sqrt(2)... why??? Can someone please enlighten me on this as well... what is this deffintion of square rooting a number never seen it ever... sure rasing a numer to some other number and rasing that number to -1, i.e. 2^(2^-1) = 2^(1/2) = sqrt(2) or simply what times waht equals two as taught in like six grade put what on earth is this deffintion???
Well hopefully someone will be able to help me as this stuff has been around for a good of chunck of time and someone has to have poundered this stuff before me especially euler when he derrived the stuff... so although there might not be many people that can hopefully someone on here will be able to help...
THANKS MILLIONS!!!!
Hi Kate,
I'm an engineer and computer programmer not a mathemetician so these proof things can be tedious and you can spend an awful lot of time going round in circles. I'm not sure why you need to prove that
sin(pi/4) = sqrt(2)/2
and you are right, it's very obvious using pythagorean theorem if you can do it geometrically.
but I can answer some of your questions:
cis(x) is an abbreviation for
cos(x) + i*sin(x)
I believe it's pronounced "sis of x" and it's just to simplify that euler formula you used above
e^(ix) = cosx + i sin(x) = cis(x)
or basically e^(ix) = cis(x)
the square root of x is defined in math as that number which when multiplied by itself results in x.
a mathematician might be a better help for your main problem. But I think even if you plug in pi/4 into euler you're just going to go round in circles. for me as an engineer trig is basically a symbolic notation for right-triangle geometry.
the geometry is much easier to work with coz like you say, you can go back to pythagoras and say ok, well a^2 + b^2 = c^2 and when you are at pi/4, both a and b are the same. So 2 * a^2 = c^2 and therefore a = b = c / sqrt(2)
now, since sine is opposite over hypot, or b/c, the sin(pi/4) = c / sqrt(2) / c or just 1/sqrt(2)
multiply top and bottom by sqrt(2) and you have your formula.
that's all I can offer. good luck!
To understand why sin(pi/4) is equal to sqrt(2)/2, you can use Euler's formula, which states that e^(ix) = cos(x) + i*sin(x). In this case, let's plug in x = pi/4:
e^(i(pi/4)) = cos(pi/4) + i*sin(pi/4).
Since cos(pi/4) = sin(pi/4) = 1/sqrt(2) (which can be derived using geometric proofs), we have:
e^(i(pi/4)) = 1/sqrt(2) + i*(1/sqrt(2)).
Now, let's simplify the numerator using Euler's formula:
e^(i(pi/4)) - e^(-i(pi/4)) = (1/sqrt(2) + i*(1/sqrt(2))) - (1/sqrt(2) - i*(1/sqrt(2))).
This simplifies to:
2i*(1/sqrt(2)) = 2i/sqrt(2) = (2i/sqrt(2)) * (sqrt(2)/sqrt(2)) = (2i*sqrt(2))/(sqrt(2)*sqrt(2)) = 2i*sqrt(2)/2 = i*sqrt(2).
So, the numerator is equal to i*sqrt(2). Dividing this by 2i, we get:
(i*sqrt(2))/(2i) = sqrt(2)/2.
Thus, sin(pi/4) is equal to sqrt(2)/2.
Regarding the term "cis", it stands for "cos + i*sin". It is a shorthand notation used to represent complex numbers in polar form. So, cis(x) is equivalent to cos(x) + i*sin(x).
As for your observation that dividing by i yields the exact value of sqrt(2), it can be explained as follows:
(i*sqrt(2))/(i) = sqrt(2).
By canceling out the i's, you're left with the value of sqrt(2).
I hope this explanation helps!