**Introduction to Graphs of Circular Functions—Domain and Range**

Each circular function relates the value of one variable to the value of another, and can be plotted as a graph in rectangular coordinates. Each of the circular functions can be “turned inside-out”; that is, the independent variable and the dependent variable can be interchanged. This gives rise to the *inverse circular functions.* In this chapter, we’ll look at the graphs of the circular functions, and also at the graphs of their inverses.

Now that you have begun to get familiar with the use of Greek letters to denote angles, we are going to go back to English letters for a while. In rectangular coordinates, the axes are usually labeled *x* (for the independent variable) and *y* (for the dependent variable). Let’s use *x* and *y* instead of *θ* and ø as the variables when graphing the circular functions. Let’s also define the terms *domain of a function* and *range of a function.*

**Domain And Range**

Suppose *f* is a function that maps (or assigns) some or all of the elements from a set *A* to some or all of the elements of a set *B.* Let *A** be the set of all elements in set *A* for which there is a corresponding element in set *B.* Then *A** is called the *domain* of *f* . Let *B** be the set of all elements in set *B* for which there is a corresponding element in set *A.* Then *B** is called the *range* of *f* .

**Graph Of Sine Function**

Figure 3-1 is a graph of the function *y* = sin *x* for values of the domain between –540° and 540° (–3π rad and 3π rad). Actually, the domain of the sine function extends over all possible values of *x;* it is the entire set of real numbers. We limit it here because our page is not infinitely wide! The range of the sine function is limited to values between, and including, –1 and 1. This curve is called a *sine wave* or *sinusoid.* It is significant in electricity, electronics, acoustics, and optics, because it represents an alternating-current (a.c.) signal with all of its energy concentrated at a single frequency.

**Fig. 3-1** . Graph of the sine function for values of *x* between –3π rad and 3π rad.

**Graph Of Cosine Function**

Figure 3-2 is a graph of the function *y* = cos *x* for values of the domain between –540° and 540° (–3π rad and 3π rad). As is the case with the sine function, the domain of the cosine function extends over the whole set of real numbers. Also like the sine function, the range of the cosine function is limited to values between, and including, –1 and 1. The shape of the *cosine wave* is exactly the same as the shape of the sine wave. Like the sine wave, the cosine wave is *sinusoidal.* The only difference is that the cosine wave is shifted horizontally in the graph by 90° (π/2 rad), or ¼ cycle, with respect to the sine wave.

**Fig. 3-2.** Graph of the cosine function for values of *x* between –3π rad and 3π rad.

**Graph Of Tangent Function**

Figure 3-3 is a graph of the function *y* = tan *x* for values of the domain between –540° and 540° (–3π rad and *3π* rad). The range of the tangent function encompasses the entire set of real numbers. But the domain does not! The function “blows up” for certain specific values of *x.* The “blow-up values” are shown as vertical, dashed lines representing *asymptotes.* For values of *x* where these asymptotes intersect the *x* axis, the function *y* = tan *x* is undefined. These values, which include all odd integral multiples of 90° (π/2 rad), are not part of the domain of the tangent function, but all other real numbers are. The term *integral multiple* means that the quantity can be multiplied by any integer, that is, any number in the set {..., –3, –2, –1, 0, 1, 2, 3, ...}.

**Fig. 3-3.** Graph of the tangent function for values of *x* between –3π rad and 3π rad.

**Graph Of Cosecant Function**

Figure 3-4 is a graph of the function *y* = csc *x* for values of the domain between –540° and 540° (–3π rad and *3π* rad). The range of the cosecant function encompasses all real numbers greater than or equal to 1, and all real numbers less than or equal to –1. The open interval representing values of *y* between, but not including, –1 and 1 is not part of the range of this function. The domain includes all real numbers except integral multiples of 180° ( *π* rad). When *x* is equal to any integral multiple of 180° ( *π* rad), the cosecant function “blows up.”

**Fig. 3-4.** Graph of the cosecant function for values of *x* between –3π rad and 3π rad.

**Graph Of Secant Function**

Figure 3-5 is a graph of the function *y* = sec *x* for values of the domain between –540° and 540° (–3π rad and *3π* rad). The range of the secant function encompasses all real numbers greater than or equal to 1, and all real numbers less than or equal to –1. Thus, the range of the secant function is the same as the range of the cosecant function. But the domain is different. It includes all real numbers except odd integral multiples of 90° *(π/2* rad). The cosecant and secant functions have the same general shape, but they are shifted by 90° (π/2 rad), or ¼ cycle, with respect to each other. This should not come as a surprise, because the cosecant and secant functions are the reciprocals of the sine and cosine functions, respectively, and the sine and cosine are horizontally displaced by ¼ cycle.

**Fig. 3-5.** Graph of the secant function for values of *x* between –3π rad and 3π rad.

**Graph Of Cotangent Function**

Figure 3-6 is a graph of the function *y* = cot *x* for values of the domain between –540° and 540° (–3π rad and 3π rad). The range of the cotangent function encompasses the entire set of real numbers. The domain skips over the integral multiples of 180° ( *π* rad). The graph of the cotangent function looks similar to that of the tangent function. The curves have the same general shape, but while the tangent function always slopes upward as you move toward the right, the cotangent always slopes downward. There is also a *phase shift* of ¼ cycle, similar to that which occurs between the cosecant and the secant functions.

**Fig. 3-6.** Graph of the cotangent function for values of *x* between –3π rad and 3π rad.

**Graphs of Circular Functions Practice Problems**

**Practice 1**

Suppose we take the unit circle, as defined in previous chapters, and cut off its bottom half, but leaving the points ( *x,y* ) = (1,0) and ( *x,y* ) = (–1,0). This produces a true mathematical function, as opposed to a mere relation, because it ensures that there is never more than one value of *y* for any value of *x.* What is the domain of this function?

**Solution 1**

You might want to draw the graph of the unit circle and erase its bottom half, placing a dot at the point (1,0) and another dot at the point (–1,0) to indicate that these points are included in the curve. The domain of this function is represented by the portion of the *x* axis for which the function is defined. It’s easy to see that this is the span of values *x* such that *x* is between –1 and 1, inclusive. Formally, if we call *A** the domain of this function, we can write this:

*A* * = { *x* : –1 ≤ *x* ≤ 1}

The colon means “such that,” and the curly brackets are set notation. So this “mathematese” statement literally reads “ *A** equals the set of all real numbers *x* such that *x* is greater than or equal to –1 and less than or equal to 1.” Sometimes a straight, vertical line is used instead of a colon to mean “such that,” so it is also acceptable to write the statement like this:

*A* * = { *x* | – 1 ≤ *x* ≤ 1}

**Practice 2**

What is the range of the function described above?

**Solution 2**

Look at the drawing you made, showing the graph of the function. The range of this function is represented by the portion of the *y* axis for which the function is defined: all the values *y* such that *y* is between 0 and 1, inclusive. Formally, if we call *B** the range of this function, we can write

*B* * = { *y* : 0 ≤ *y* ≤ 1}

**Practice 3**

The domain of the sine function is the same as the domain of the cosine function. In addition, the ranges of the two functions are the same. How can this be true, and yet the two functions are not identical?

**Solution 3**

The difference, as you can see by comparing the graphs of the two functions, is that the curves are displaced along the *x* axis by 90° (π/2 rad). In general, the cosine of a number is not the same as the sine of that number, although there are certain specific instances in which the two functions have the same value.

**Practice 4**

Draw a graph that shows the specific points where sin *x* = cos *x.*

**Solution 4**

This can be done by superimposing the sine wave and the cosine wave on the same set of coordinates, as shown in Fig. 3-7. The functions attain the same value where the curves intersect.

**Fig. 3-7** . Illustration for Solution 4, showing points where the sine and cosine functions attain the same *y* value.

Practice Problems for these concepts can be found at: Graphs and Inverse Practice Test