### Domain, Range, and Signs of Functions

While the trigonometric functions may seem quite different from other functions you have worked with, they are in fact just like any other function. We can think of a trig function in terms of "input" and "output." The input is always an angle. The output is a ratio of sides of a triangle. If you think about the trig functions in this way, you can define the domain and range of each function.

Let's first consider the sine and cosine functions. The input of each of these functions is always an angle, and as you learned in the previous chapter, these angles can take on any real number value. Therefore the sine and cosine function have the same domain, the set of all real numbers, . We can determine the range of the functions if we think about the fact that the sine of an angle is the coordinate of the point where the terminal side of the angle intersects the unit circle. The cosine is the coordinate of that point. Now recall that in the unit circle, we defined the trig functions in terms of a triangle with hypotenuse .

In this right triangle, and are the lengths of the legs of the triangle, which must have lengths less than , the length of the hypotenuse. Therefore the ranges of the sine and cosine function do not include values greater than one. The ranges do, however, contain negative values. Any angle whose terminal side is in the third or fourth quadrant will have a negative coordinate, and any angle whose terminal side is in the second or third quadrant will have a negative coordinate.

In either case, the minimum value is . For example, and . Therefore the sine and cosine function both have range from to .

The table below summarizes the domains and ranges of these functions:

 Domain Range Sine Cosine

Knowing the domain and range of the cosine and sine function can help us determine the domain and range of the secant and cosecant function. First consider the sine and cosecant functions, which as we showed above, are reciprocals. The cosecant function will be defined as long as the sine value is not . Therefore the domain of the cosecant function excludes all angles with sine value , which are , etc.

In Chapter 2 you will analyze the graphs of these functions, which will help you see why the reciprocal relationship results in a particular range for the cosecant function. Here we will state this range, and in the review questions you will explore values of the sine and cosecant function or order to begin to verify this range, as well as the domain and range of the secant function.

 Domain Range Cosecant or Secant or

Now let's consider the tangent and cotangent functions. The tangent function is defined as . Therefore the domain of this function excludes angles for which the ordered pair has an coordinate of : , , etc. The cotangent function is defined as , so this function's domain will exclude angles for which the ordered pair has a coordinate of : , , , etc. As you will learn in chapter 3 when you study the graphs of these functions, there are no restrictions on the ranges.

In either case, the minimum value is . For example, and . Therefore the sine and cosine function both have range from to .

The table below summarizes the domains and ranges of these functions:

 Domain Range Sine Cosine

Knowing the domain and range of the cosine and sine function can help us determine the domain and range of the secant and cosecant function. First consider the sine and cosecant functions, which as we showed above, are reciprocals. The cosecant function will be defined as long as the sine value is not . Therefore the domain of the cosecant function excludes all angles with sine value , which are , etc.

In Chapter 2 you will analyze the graphs of these functions, which will help you see why the reciprocal relationship results in a particular range for the cosecant function. Here we will state this range, and in the review questions you will explore values of the sine and cosecant function or order to begin to verify this range, as well as the domain and range of the secant function.

 Domain Range Cosecant or Secant or

Now let's consider the tangent and cotangent functions. The tangent function is defined as . Therefore the domain of this function excludes angles for which the ordered pair has an coordinate of : , , etc. The cotangent function is defined as , so this function's domain will exclude angles for which the ordered pair has a coordinate of : , , , etc. As you will learn in chapter 3 when you study the graphs of these functions, there are no restrictions on the ranges.

 Function Domain Range Tangent Cotangent