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

by tatiana_eggs
Tags: imaginary, real, sin4
 P: 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?
 Quote by tatiana_eggs 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?
Now use $e^{3+ 4i}= e^3cos(4)+ i e^3sin(4)$, etc.