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

[tatiana_eggs]

Homework Statement

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

Homework 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

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?

 

[HallsofIvy]

Homework Statement

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

Homework 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

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?

Now use e^{3+ 4i}= e^3cos(4)+ i e^3sin(4), etc.

 

[tatiana_eggs]

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