There are several definitions of the function #sinx#. I will give three of the most common ones below.
(i) Probably the simplest is in relation to any right #triangle ABC# where #angle C# is a right angle (#90^0 -= pi/2 rad#). The sides of #triangle ABC# are #a, b, c# opposite their respective angle. Here side #c# is called the hypotenuse as it is opposite the right angle.
#sin theta# is defined as the ratio: #"opposite / hypotenuse"#.
So, in the case above #sin A = a/c; sin B =b/c#
(ii) Consider the unit circle centred at the origin #(O)#. A point #P (x,y)# moves around the circumference in a counter clockwise direction from the point #(1,0)#. Let the angle formed between #OP# and the positive #x-#axis be #theta#
#sin theta# is then defined as the #y-#component of #P#
NB: This definition is equivalent to (i) above for #theta in (0, pi/2)# and generalises the function for #theta in [0,2pi]# and, by extension, to all real numbers.
(iii) #sinx# is continuous and differentiable for all real numbers - #(forall x in RR)#
Hence, it can be defined by a Taylor series at #x=0#. This series expansion is:
#sinx = x-x^3/(3!)+x^5/(5!)-x^7/(7!)+....#
#= sum_"n=0"^oo (-1)^n/((2n+1)!) x^(2n+1)#
Finally, to "represent" #y=sinx# as requested in the question, the graph of #y=sinx# is shown below.
graph{sinx [-6.244, 6.244, -3.12, 3.124]}
