Final answer:
The equation α = 1 + 0.25sin(2α) is simplified using a small angle approximation, leading to the rough solution α = 2. More accurately, the solution is found where the curves y = α and y = 1 + 0.25sin(2α) intersect, which requires graphical or numerical methods.
Step-by-step explanation:
To show that the equation x = 1 + 0.25sin(2x) has a unique solution α, we need to examine its behavior. If we start by assuming that x is relatively small, we can approximate sin(2x) as 2x, due to the small angle approximation, where sine of the angle is approximately equal to the angle itself when the angle is small (in radians).
Our simplified equation becomes x = 1 + 0.25(2x), or x = 1 + 0.5x. Rearranging gives 0.5x = 1 or x = 2. This suggests that a solution exists around x = 2.
However, this is a transcendental equation, meaning the exact value requires iterative methods or graphical analysis to determine. While the assumption that x is small was used for the simplification, it is more accurate to state that a solution exists by checking the intersection of the curves y = x and y = 1 + 0.25sin(2x), which can be done graphically or numerically. The unique solution happens where these two curves intersect exactly once within a certain interval. The continuity and bounded nature of the sine function suggest the presence of at least one solution.