How do you find the domain and range of sine, cosine, and tangent?
1 Answer
The domain and range of trigonometric functions are determined directly from the definition of these functions.
Let's start from the definition.
Trigonometric functions are defined using a unit circle on a coordinate plane  a circle of a radius
Consider a point
The value of this angle can be positive (if we go counterclockwise from
Each value of an angle from

Function
#y=sin(x)# is defined as the ordinate (#Y# coordinate) of a point on a unit circle that corresponds to an angle of#x# radians. Therefore, the domain of this function is all real numbers from#oo# to#+oo# . The range is from#1# to#+1# since this is an ordinate of a point on a unit circle. 
Function
#y=cos(x)# is defined as the abscissa (#X# coordinate) of a point on a unit circle that corresponds to an angle of#x# radians. Therefore, the domain of this function is all real numbers from#oo# to#+oo# . The range is from#1# to#+1# since this is an abscissa of a point on a unit circle. 
Function
#y=tan(x)# is defined as#sin(x)/cos(x)# . The domain of this function is all real numbers except those where#cos(x)=0# , that is all angles except those that correspond to points#(0,1)# and#(0,1)# . These angles where#y=tan(x)# is undefined are#pi/2 + pi*N# radians, where#N#  any integer number. The range is, obviously, all real numbers from#oo# to#+oo# .
Of special interest might be the graphs of these functions. You can refer to a series of lectures on Unizor dedicated to detailed analysis of these functions, their graphs and behavior.