Sure, let's find all angles θ in the interval [0,360) for which the sine function equals 0.9272.
1. First, we notice that the sine function equals any value in the range [-1, 1] in two different intervals per cycle. This is because the sine function is positive in the first and second quadrants, then negative in the third and fourth quadrants.
2. We are dealing with an interval of [0,360). The sine function reaches a range from -1 to 1 during that interval. Since the value we are looking for, 0.9272, is positive, we can keep searching for solutions in the first and second quadrants, i.e., the degree range [0, 180].
3. Check all angles in the degree range [0, 180] to find approximate solutions. The calculation goes as follows.
4. Starting from 0 degree, we check each angle consecutively. The approximated angles that satisfy sin(θ) = 0.9272 are found to be [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123].
5. We finished our search on 180 degree. All the degrees from the interval [0, 360) that satisfy the equation sin(θ) = 0.9272 were found within the degree range [57, 123], giving the aforementioned range of solutions.
And that is how you find all approximated solutions to the given equation within the specified degree range [0, 360).