How do you find the domain and range of sine, cosine, and tangent?

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 #1# centered at the origin of coordinates #O#.

Consider a point #A# on this circle and an angle from the positive direction of the #X#-axis (ray #OX#) to a ray #OA# that connects a center of a unit circle with our point #A#. This angle can be measured in degrees or, more commonly in the analysis of trigonometric functions, in radians.

The value of this angle can be positive (if we go counterclockwise from #OX# to #OA#) or negative (going clockwise). It can be greater by absolute value than a full angle (the one of #360# degrees or #2pi# radians), in which case the position of a point #A# is determined by circulating around the center of a unit circle more than once.

Each value of an angle from #-oo# to #+oo# (in degrees or, more preferably, radians) corresponds to a position of a point on the unit circle. For each such angle the values of trigonometric functions are defined as follows.

1. 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.

2. 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.

3. 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.