Graphing Sine and Cosine
Key Questions

The domain of a function
#f(x)# are all the values of#x# for which#f(x)# is valid.The range of a function
#f(x)# are all the values which#f(x)# can take on.#sin(x)# is defined for all real values of#x# , so it's domain is all real numbers.However, the value of
#sin(x)# , its range , is restricted to the closed interval [1, +1]. (Based on definition of#sin(x)# .) 
The maximum value of the function
#cos(x)# is#1# .This result can be easily obtained using differential calculus.
First, recall that for a function
#f(x)# to have a local maximum at a point#x_0# of it's domain it is necessary (but not sufficient) that#f^prime(x_0)=0# . Additionally, if#f^((2)) (x_0)<0# (the second derivative of f at the point#x_0# is negative) we have a local maximum.For the function
#cos(x)# :#d/dx cos(x)=sin(x)# #d^2/dx^2 cos(x)=cos(x)# The function
#sin(x)# has roots at points of the form#x=n pi# , where#n# is an integer (positive or negative).The function
#cos(x)# is negative for points of the form#x= (2n+1) pi# (odd multiples of#pi# ) and positive for points of the form#2n pi# (even multiples of#pi# ).Therefore, the function
#cos(x)# has all it's maximums at the points of the form#x=(2n+1)pi# , where it takes the value#1# . 
Since the function passes from the origin, in fact
#sin0=0# ,
the yintercept is#0# .graph{sinx [10, 10, 5, 5]}