# Find the real and imaginary part of sin(4+3i)

1. ### tatiana_eggs

70
1. The problem statement, all variables and given/known data

Find the real and imaginary part of sin(4+3i)

2. Relevant equations

sinx = $$\frac{e^z - e^(-z)}{2i}$$

cosx = $$\frac{e^z + e^(-z)}{2}$$

sin(iy) = i$$\frac{e^y - e^(-y)}{2}$$

cos(iy) = $$\frac{e^y + e^(-y)}{2}$$

various trig identities

3. The attempt at a solution

So I used sin(x+y) trig identity and got
sin4*cos3i + sin3i*cos4

I turned them all into exponents using the appropriate equations stated in (2).

I got to a point where nothing is really calculable by hand/head. Is there an easier way to do this or does the calculator need to be used at a certain point to calculate the real part(terms grouped w/o i) and the imaginary part (terms grouped with i).

If so, then I guess I need help getting the terms grouped together to calculate the real and imaginary parts.

Where I am stuck is at:

$$\frac{e^{3+4i}+e^{-3+4i}-e^{3-4i}+e^{-3-4i}}{4i}$$ +
$$\frac{e^{3+4i}-e^{-3+4i}+e^{3-4i}-e^{-3-4i}}{4}$$

(the two fractions should be added together)

Now what should I do with all these lovely exponents? Should I have even gone this route?

2. ### HallsofIvy

40,918
Staff Emeritus
Now use $e^{3+ 4i}= e^3cos(4)+ i e^3sin(4)$, etc.

3. ### tatiana_eggs

70
That was just the hint I needed, Halls. Thanks! Finally got it.