# 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.