I don't think there is a reasonable way to calculate trigonometric functions by hand. The computer solves a number of terms in an infinite series to approximate it. Back in the day there used to be books of tables, that had a lits of trigonometric functions logarithms, and so on for different numbers. I guess you could also use a slide rule. As a side note, there is an old timey isekai by an American author, I forget the name, were the magician has to give up his magic to get a book of tables and it's considered a good deal.
The closer thing to solving trigonometric functions by hand that can be taught in grade school is the unit circle. Incidentally when you buy a ruler set, it comes with 2 triangles and a compass, the 2 triangles have 30 and 45 degree angles Wich are the ones used for the unit circle.