Final answer:
To solve the trigonometric equation 2 sin(1/2 x) = 1, first divide by 2 to get sin(1/2 x) = 1/2, then find the corresponding angles (30 and 150 degrees) and multiply by 2. The solutions within the range 0 < x ≤ 12 are x = 60 degrees and x = 300 degrees.
Step-by-step explanation:
To solve the trigonometric equation 2 sin(1/2 x) = 1 for all values 0 < x ≤ 12, we first isolate the sine term.
- Divide both sides by 2 to get sin(1/2 x) = 1/2.
- Find the angle whose sine is 1/2. We know that sin(π/6) or sin(30 degrees) = 1/2.
- Since the sine function is periodic, we'll have two solutions in the range of 0 to 2π for 1/2 x, which are π/6 and 5π/6 (or 30 and 150 degrees).
- Now, we multiply these angles by 2 to solve for x, getting x = π/3 and 5π/3 (or x = 60 and 300 degrees).
- Convert these solutions to degrees if necessary and examine additional solutions in the range of 0 to 12 by considering the periodicity of the sine function. The sine function has a period of 2π, so we will look for solutions of the form x = 60 + n*360 and x = 300 + n*360, where n is an integer such that the solution remains within the range.
Therefore, the solutions to the given trigonometric equation within the specified range are x = 60 degrees (or π/3 radians) and x = 300 degrees (or 5π/3 radians).