Find the real and Imaginary parts of sin(3+i)

1. Dec 9, 2013

ChloeFoulkes

1. The problem statement, all variables and given/known data
Find the real and Imaginary parts of sin(3+i)

2. Relevant equations
sin(x+y)= sinxcosy+sinycosx

3. The attempt at a solution
I think im right in saying that you use the sine addition formula but then i get stuck from there.
Is it something to do with exponential form?

2. Dec 9, 2013

ShayanJ

You just lack the formulas $\sin{(ix)}=i\sinh{x}$ and $\cos{(ix)}=\cosh{x}$.

3. Dec 9, 2013

Saitama

You can use the following

$$\sin x=\frac{e^{ix}-e^{-ix}}{2}$$

4. Dec 9, 2013

ChloeFoulkes

I get:1/4i(e-1+e1)(e3i-e-3i)+1/4i(e3i+e-3i)(e-1-e1)

now would i sum the real and imaginary parts?

5. Dec 9, 2013

Saitama

I did not check the simplification. You can use $e^{ix}=\cos x+i\sin x$. :)

6. Dec 9, 2013

Seydlitz

Extending on the suggestion, you can hence construct a complex double angle identity of $\sin(x+iy)$ with hyperbolic function. If you have constructed that you can just plug in the values of real x and y. I suggest you do the same with other trig function with $z=(x+iy)$ as the argument, it helps a lot. Either that or you can also use the exponential form Pranav-Arora mentioned,

which should actually be:
$$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$

7. Dec 9, 2013

Saitama

Ah yes, very sorry.

8. Dec 9, 2013

ChloeFoulkes

im gonna use the exponential method, only because it rings a bell with something we did in a lecture:)
ive now got:
1/4i(e-1+e1)(cos3+isin3-cos(-3)+isin(-3))+1/4i(cos3+isin3+cos(-3)+isin(-3))(e-1-e1)

all the trig seems to cancel out?
so im left with 1/4i(e-1+e1)+1/4i(e-1-e1)

I still dont understand how this gives me the real and imaginary parts? or am i being abit 'maths blind'

9. Dec 9, 2013

PeroK

You've gone wrong. The trig shouldn't cancel out. You're better expanding the sin equation as was previously suggested to get:

$sin(x + iy) = sin(x)cosh(y) + icos(x)sinh(y)$

This is much more useful than the expoential form when you need to calculate the sine of a complex number.

10. Dec 9, 2013

vela

Staff Emeritus
I didn't check carefully, but I think you made some sign errors. Also, you seem to think that $\cos (-x) = -\cos x$ and $\sin (-x) = -\sin x$. Only one of those is true. You need to show your work rather than just posting your answers so we can see where you're making mistakes.

By the way, you've made a bit more work for yourself than is necessary by using the identity first and then converting to exponentials. Just go straight to the exponential form:
$$\sin (3+i) = \frac{e^{i(3+i)}-e^{-i(3+i)}}{2i}.$$
To find the real and imaginary parts, you eventually want to express the answer in the form x+iy, where x and y are real. That means, in particular, you're going to need to get the $i$ out of the denominator.