Why are the solutions different?

[ijabbott]

On the Prime, you just need to enter HMS/DMS values in sexagesimal format instead of the decimal format that you're used to. It's no big deal as there is a Math Template to enter sexagesimal numbers. Admittedly, it's not as convenient as entering sexagesimal numbers on Casio calculators.

You also don't need to use the HMS→ function inside the SIN function to convert the sexagesimal number to decimal. The SIN function can work on the sexagesimal number directly, treating it as DMS. For example, SIN(40°35’00”) ≈ 0.650553326665.