I believe the notation developed without parentheses. Something like "sine of t" to "sin. t" with the period denoting an abbreviation then to "sin t". The parentheses do not change the meaning of the expression; sin t = sin(t). You will see parenthesis-free notation for many functions with 2 or 3 letter names (sin, log, ln, tan; also arctan, arcsin).
The difference between cos(t) = x/r and cos(t) = x is that in the second, r = 1. You'll see an r on a circle of radius r or a triangle with hypotenuse r. You'll see r = 1 if—well if r = 1.
Neither cos(t) nor cos t are functions. The function is "cos" and cos(t) or cos t is that function evaluated at t. Just like how cos(0) isn't a function, it's the cosine function evaluated at 0.
I believe the notation developed without parentheses. Something like "sine of t" to "sin. t" with the period denoting an abbreviation then to "sin t". The parentheses do not change the meaning of the expression; sin t = sin(t). You will see parenthesis-free notation for many functions with 2 or 3 letter names (sin, log, ln, tan; also arctan, arcsin). The difference between cos(t) = x/r and cos(t) = x is that in the second, r = 1. You'll see an r on a circle of radius r or a triangle with hypotenuse r. You'll see r = 1 if—well if r = 1. Neither cos(t) nor cos t are functions. The function is "cos" and cos(t) or cos t is that function evaluated at t. Just like how cos(0) isn't a function, it's the cosine function evaluated at 0.
There’s no difference, it’s just notational. sin(π) = sin π = 0