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Spud

Guest

Sine and Cosine of i (imaginary unit)

The following is my working for the Sine and Cosine of the imaginary unit, [MATH]i[/MATH].

[MATH]e^{ix}=cos(x)+i(sin(x))[/MATH]
Substituting [MATH]x=i[/MATH] gives
[MATH]e^{-1}=cos(i)+i(sin(i))[/MATH]
Squaring both sides
[MATH]e^{-2}=(cos(i))^2-(sin(i))^2+2i(sin(i)cos(i))[/MATH]
[MATH]e^{-2}=(1-2((sin(i))^2)+2i(sin(i)cos(i))[/MATH]
Rearranging the original equation in terms of [MATH]cos(i)[/MATH] gives
[MATH]cos(i)=e^{-1}-i(sin(i))[/MATH]
Substituing this back into [MATH]e^{-2}=(1-2((sin(i))^2)+2i(sin(i)cos(i)) [/MATH]gives
[MATH]e^{-2}=(1-2((sin(i))^2)+2i(sin(i))(e^{-1}-i(sin(i)))[/MATH]
[MATH]e^{-2}=(1-2((sin(i))^2)+2i(sin(i))(e^{-1}) -2(i^2)((sin(i))^2)[/MATH]
[MATH]e^{-2}=1+2i(sin(i))(e^{-1})[/MATH]
Rearranging for sin(i) gives
[MATH](e^{-2}-1)/2i(e^{-1})=sin(i)[/MATH]
[MATH]sin(i)=(1-e^2)/2ei[/MATH]
Then to find Cos(i): Substituting back into the original equation gives
[MATH]cos(i)=e^{-1}-(i(1-e^2))/(2ei)
cos(i)=(1-e^2)/2e[/MATH]

The problem is that all the sources I can find for the sine and cosine of [MATH]i[/MATH] on the internet say
[MATH]sin(i) = ((e-e^{-1})/2)i[/MATH] or [MATH]((e^2-1)/2e)i[/MATH] and
[MATH]cos(i)=(e+e^{-1})/2[/MATH] or ([MATH]e^2+1)/2e [/MATH].

As you can see my answers are very close to the others, I just can't figure out where the difference in signs comes from...

also, is there any difference if the [MATH]i[/MATH] is put outside the fraction like they have, or on the bottom like I work it out to be?

If I've made any typos or not made anything clear enough, just point it out and I'll change the original post.

Thanks, Spud.

Spud

Guest

Re: Sine and Cosine of i (imaginary unit)

Sorry, looks like i should've put math instead of MATH. I'll try again..

The following is my working for the Sine and Cosine of the imaginary unit,

.

Substituting

gives

Squaring both sides

Rearranging the original equation in terms of

gives

Substituing this back into

gives

Rearranging for sin(i) gives

Then to find Cos(i): Substituting back into the original equation gives

The problem is that all the sources I can find for the sine and cosine of

on the internet say

or

and

or (

.

As you can see my answers are very close to the others, I just can't figure out where the difference in signs comes from...

also, is there any difference if the

is put outside the fraction like they have, or on the bottom like I work it to

be?

If I've made any typos or not made anything clear enough, just point it out and I'll change the original post.

Thanks, Spud.

Spud

Guest

Re: Sine and Cosine of i (imaginary unit)

Edit: Found my mistake in working for Cos(i), it is right after all. But can't yet see what;s wrong with sin(i).

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Sine and Cosine of i (imaginary unit)

You have

and

Hence