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 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
To find the smallest positive number t such that e^sin(t) = 1/2, we need to solve the equation for sin(t) first and then find the corresponding value of t.
Step 1: Take the natural logarithm (ln) of both sides to eliminate the exponential function:
ln(e^sin(t)) = ln(1/2).
Step 2: Apply the logarithmic property ln(e^x) = x:
sin(t) = ln(1/2).
Step 3: Use the inverse sine function (sin^(-1)) to isolate t:
t = sin^(-1)(ln(1/2)).
Step 4: Evaluate t using a calculator:
t ≈ 0.7302 (rounded to four decimal places).
Therefore, the smallest positive number t that satisfies e^sin(t) = 1/2 is approximately 0.7302.