The table shows the height, in feet, of the tide over time, in hours. i know the answer is that it should be cosine instead of sine, but i don't how to determine that. i'd like to know so i could solve questions like these in the future.

For this case we have the following function:
 f (x) = - 4sine ((pi / 6) x) + 5
 We evaluate the function for x = 0
 f (0) = - 4sine ((pi / 6) 0) + 5
 f (0) = -4 * (0) + 5
 f (0) = 5
 We note that the result does not correspond to the table.
 Let's rewrite the function:
 f (x) = - 4cosine ((pi / 6) x) + 5
 We evaluate the function for x = 0
 f (0) = - 4cosine ((pi / 6) 0) + 5
 f (0) = -4 * (1) + 5
 f (0) = 1
 The result corresponds to the table
 
 The student's mistake is to use the sine function instead of using the cosine function.
 The correct function is:
 f (x) = - 4cosine ((pi / 6) x) + 5