You say "I know that function sin is periodic at ever 2*pi". That is exactly what "periodic" means: sin(x+ 2pi)= sin(x). The function value repeats "periodically"- at regular intervals.

You also say "however the argument of sin is not the same(x != x+(2*pi)) we are getting the same function/answer?". The fact that \(\displaystyle x\ne y\) does NOT necessarily mean that f(x) cannot be equal to f(y). Look at the very simple, *constant* function "f(x)= 3 for all x". That is a perfectly valid, though very simple, function. Look at \(\displaystyle y= x^2\). f(-3)= 9 and f(3)= 9. The fact that the two "x" values are different doesn't mean the function values must be different.

Finally, exactly what **definition **of "sine" and "cosine" were you given? The definitions in terms of right triangles won't work since an angle, in a right triangle must be positive and can't be larger than pi/2 radians so "x+ 2pi" wouldn't even make sense. Most common is the "circle definition" (and some texts use the phrase "circular functions" rather than "trigonometric functions"): Draw the unit circle on a coordinate system-center at the origin, radius 1. Starting from the point (1, 0) measure a distance "t" around the circumference of the circle. The (x, y) coordinates of the end point give sine and cosine- cos(t) is the x coordinate, sin(t) is the y coordinate.

For example, since the circle has radius 1, it has circumference 2pi(1)= 2pi. pi/2 is 1/4 of that. If we start at (1, 0) and measure distance pi/2 around the circumference we go 1/4 of the way, ending at (0, 1). So cos(pi/2)= 0, sin(pi/2)= 1. If instead we measure distance pi, we go half way around the circle, from (1, 0) to (-1, 0). cos(pi)= -1, sin(pi)= 0. A little harder is sin(pi/4) and cos(pi/4). Since that is half of pi/2, we wind up half way between (1, 0) and (0, 1). By symmetry we are on the line y= x. The equation of the unit circle is x^2+ y^2= 1. Since y= x, we have x^2+ x^2= 2x^2= 1 so x^2= 1/2 and x= sqrt(1/2)= sqrt(2)/2 (we are still in the first quadrant so x and y are both positive). That is, cos(pi/4)= sin(pi/4)= sqrt(2)/2.

But, again, the entire circle has circumference 2pi. If I measure a distance around the circle x+ 2pi, I go from (1, 0) to (cos(x), sin(x)) and then on another 2pi. I have measured a distance x+ 2pi so "by definition" I must end at the point (cos(x+ 2pi), sin(x+ 2pi). But that last 2pi takes me exactly once around the circle so I come right back to (cos(x), sin(x)). So cos(x+ 2pi)= cos(x), sin(x+ 2pi)=sin(x).