Question

How do you find values for the six trigonometric functions for angles of rotation?

Answer 1

Any angle of rotation `theta` can be represented by a point `A` on a unit circle with a center at the origin of coordinates `O` and radius `1`. The angle is measured counterclockwise from the positive direction of the X-axis to a line from `O` to `A`, so `angle XOA=theta` with `|OA|=1`. Thus, an angle of `90^0` is represented by a point with coordinates `(0,1)`, an angle of `270^0` is represented by a point `(0,-1)` etc.

Then, by definition, if point `A` has coordinates `(A_x,A_y)`,
`sin(theta)=A_y`
`cos(theta)=A_x`
`tan(theta)=A_y/A_x` (for `A_x!=0`)
`sec(theta)=1/A_x` (for `A_x!=0`)
`csc(theta)=1/A_y` (for `A_y!=0`)

The above are definitions of trigonometric functions for any angles. The typical geometric definition of trigonometric functions using the right triangles is not general enough, while the above definitions work for all angles and, in case of acute angles in the right triangles, are identical to geometric definition.

I might suggest to study trigonometry at Unizor - Trigonometry. The site has a very detailed explanation of properties of trigonometric functions based on the above definition.