Final answer:
The equation cosine(x/2) = cosine(x) + 1 has no solutions in the interval 0° ≤ x < 360° because cosine values are restricted to the range [-1, 1], making the right-hand side of the equation impossible to achieve.
Step-by-step explanation:
The student's question asks to find solutions for the equation cosine(x/2) = cosine(x) + 1 within the interval 0° ≤ x < 360°. Firstly, we should observe that the cosine function has a range from -1 to 1, meaning that the maximum value of cosine(x) can be 1. Therefore, the equation cosine(x) + 1 would yield a maximum of 2, which is outside the possible range for a cosine function. This suggests there are no solutions because cosine(x/2) can never equal a value greater than 1, yet the equation implies that it must equal a value which could be up to 2.
To evaluate this, recall the basic properties of cosine functions. The fact that cosine(x) could be at a maximum of 1 when x is 0° indicates that at this point, cosine(x/2) would also be 1 since 0°/2 is still 0°. However, the addition of 1 to cosine(x) makes it impossible for the equation to hold, as the cosine of an angle cannot be greater than 1.