The solutions for the equation sin(θ + π/4) = 1/2 in the interval [0, 2π] are:
θ1 = 0.523 radians + 2πn, where n is any non-negative integer.
θ2 = 1.893 radians + 2πn, where n is any non-negative integer.
θ3 = 7.369 radians + 2πn, where n is any non-negative integer.
1. Solve for θ + π/4:
Take the arcsine (sin^-1) of both sides:
θ + π/4 = arcsin(1/2)
Arcsine has two solutions in the interval [0, 2π], because sine is periodic:
θ + π/4 = π/6 + 2πn or θ + π/4 = 5π/6 + 2πn
where n is any integer.
2. Solve for θ:
Subtract π/4 from both sides in each equation:
θ = π/6 + 2πn - π/4 or θ = 5π/6 + 2πn - π/4
Simplify:
θ = -π/12 + 2πn or θ = 7π/12 + 2πn
3. Find solutions in the interval [0, 2π]:
For the first solution, substitute n = 0, 1, 2, ... until θ is greater than or equal to 2π:
θ1(0) = -π/12 = 0.523 radians
θ1(1) = -π/12 + 2π = 5.999 radians
θ1(2) = -π/12 + 4π = 11.471 radians
The first solution that falls within the interval [0, 2π] is θ1(0) = 0.523 radians.
For the second solution, substitute n = 0, 1, 2, ... until θ is greater than or equal to 2π:
θ2(0) = 7π/12 = 1.893 radians
θ2(1) = 7π/12 + 2π = 7.369 radians
θ2(2) = 7π/12 + 4π = 12.845 radians
Both θ2(0) = 1.893 radians and θ2(1) = 7.369 radians fall within the interval [0, 2π].