No, No, No.
The fact is that most applications are sine and cosine have nothing to do with triangles. The really important thing about sine and cosine are that they are the "ideal" periodic (repeating) functions and can be used to model periodic phenomena.
The standard definition of sine and cosine is this:
Given the unit circle in some coordinate system, start at (1,0) and measure around the circumference of the circle a distance t (counterclockwise if t>0, clockwise if t<0).
The coordinates of the end point are, by definition, (cos(x), sin(t)).
Notice I said "by definition". You don't need to have any triangles or, indeed, any angles, involved in that at all. You can, of course, draw a right triangle with hypotenuse a radius of the circle so that x and y are the lengths of the legs and determine that x= cos(t)= "opposite side over hypotenuse" and y= sin(t)= "near side over hypotenuse" but those are results of the definition in terms of a circle.
Notice also that t is NOT an angle. It is a length measurement and so its units are given by the coordinate system. Unfortunately, calculators are designed by engineers rather than mathematicians and engineers tend to think of sine and cosine in terms of angles (which is why they talk about things like "phase angle" when there are no angles involved at all!). In order to make the two definitions work together, we have to interpret the "angle" as in RADIANS, not degrees. Except in specific applications where angles are measured in degrees, you should always work in radians.
