Question: How can I convert exponential functions to sine and cosine functions?

Dear All,

I have the following algebraic function. In vibrations, it is common to write the functions exp(alpha[i]*I*t) as separate terms of cos(alpha[i]*t) and sin(alpha[i]*t). While exp(-beta[i]*t) remains without converting to sinh and cosh. How can I find the solution given for T(t) as the sum of terms C1[i]*sin(alpha[i]*t)*exp(-beta[i]*t) and C2[i]*cos(alpha[i]*t)*exp(-beta[i]*t) with non-complex (real) coefficients C1[i] and C2[i].

Can anyone help me to achieve my goal in the following expression?

T(t):=(1.450761945*10^(-11) + (3.836655196*10^(-14))*I)*exp((-0.5000000000 + 222.6866468*I)*t) + (-3.770333746*10^(-9) + (2.179000257*10^(-6))*I)*exp((-0.5000000000 - 924.5904413*I)*t) + (-2.584086158*10^(-12) + (4.273321932*10^(-13))*I)*exp((-0.5000000000 + 326.7549627*I)*t) + (1.986287340*10^(-9) + (1.330623218*10^(-11))*I)*exp((-0.5000000000 - 74.63720909*I)*t) + (-5.980910367*10^(-12) + (5.816480027*10^(-11))*I)*exp((-0.5000000000 - 453.7574402*I)*t) + (1.450761945*10^(-11) - (3.836655196*10^(-14))*I)*exp((-0.5000000000 - 222.6866468*I)*t) + (8.923968224*10^(-10) - (8.844466162*10^(-9))*I)*exp((-0.5000000000 + 637.9999953*I)*t) - (1.217986141*10^(-10) + (4.431771836*10^(-13))*I)*exp((-0.5000000000 - 138.7904660*I)*t) + (-1.217986141*10^(-10) + (4.431771836*10^(-13))*I)*exp((-0.5000000000 + 138.7904660*I)*t) + (0.0002537882980 + 0.00002277791755*I)*exp((-0.5000000000 - 5.570928456*I)*t) - (3.770333746*10^(-9) + (2.179000257*10^(-6))*I)*exp((-0.5000000000 + 924.5904413*I)*t) + (-0.0001618723219 + 0.01288449595*I)*exp((-0.5000000000 - 1638.001654*I)*t) + (8.923968224*10^(-10) + (8.844466162*10^(-9))*I)*exp((-0.5000000000 - 637.9999953*I)*t) + (-1.153529195*10^(-7) + (1.908444485*10^(-9))*I)*exp((-0.5000000000 + 30.22171212*I)*t) - (1.153529195*10^(-7) + (1.908444485*10^(-9))*I)*exp((-0.5000000000 - 30.22171212*I)*t) - (0.0001618723219 + 0.01288449595*I)*exp((-0.5000000000 + 1638.001654*I)*t) + (1.986287340*10^(-9) - (1.330623218*10^(-11))*I)*exp((-0.5000000000 + 74.63720909*I)*t) + (0.0002537882980 - 0.00002277791755*I)*exp((-0.5000000000 + 5.570928456*I)*t) - (5.980910367*10^(-12) + (5.816480027*10^(-11))*I)*exp((-0.5000000000 + 453.7574402*I)*t) - (2.584086158*10^(-12) + (4.273321932*10^(-13))*I)*exp((-0.5000000000 - 326.7549627*I)*t);