Final answer:
By examining the ranges of the sine and cosine functions, we determine that the given equation 2sinx = 2 + cos3x has no solutions in the interval 0≤x≤360 because the only possible solution would require cos3x to be zero, which does not happen when sinx equals 1.
Step-by-step explanation:
To solve the equation 2sinx = 2 + cos3x in the interval 0≤x≤360, we can first observe the ranges of the functions involved. The sine function, sinx, varies between -1 and 1, so 2sinx will vary between -2 and 2. On the other side, the cosine function, cos3x, also varies between -1 and 1. For the equation to hold true, the left side must equal the right side, therefore, 2 plus something that varies between -1 and 1 must be equal to something that varies between -2 and 2.
This can only happen if the cosine term, cos3x, is zero, which means 2sinx must equal 2. This simplifies down to sinx = 1. We know that sinx = 1 only at x = 90°. Checking if cos3x is zero at this point, we see that 3x must be 90° or 270°, but since x = 90°, this does not hold true (as 3*90° does not equal 90° or 270°). Therefore, the equation has no solutions within the specified interval.