Find the smallest positive number t such that e^sin(t)=1/2

Answer 1

take ln of both sides

ln(e^sin(t) = ln(1/2)
sin(t) (lne) = ln 1 - ln 2 , but lne = 1 and ln 1 = 0

sin(t) = -ln2 = -.69314718
t must be in quads III or IV
angle in standard position is 43.9° or .7658 radians

in degrees,
t = 180+43.9 = 223.9° or 360-43.9 = 316.1°

in radians
t = π+.7658 = 3.907 radians or 2π-.7658 = 5.517 radians

check one of them
t = 316.1°
e^sin 316.1° = .4998 , not bad

so the smallest value of t = 3.907