Precalc/Trig
posted by Liz .
Find the smallest positive number t such that e^sin(t)=1/2

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 36043.9 = 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