I think the answer is potassium
At my beginning there's a vessel,
The word pot
You can slash me with a chisel.
Refers to Minecraft, which has a potassium block. (I've never played Minecraft so I'm basing this off Googling)
I belong to the first from a total of four,
First column, fourth row of the periodic table
There's nothing to add more ...
Don't know this one
Hint 1: The Windowmaker
Microsoft, which makes Windows and Minecraft
Hint 2: Blocks
Hint 3: sin, cos, tan are ........ in nature
Periodic, as in the periodic table