**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.”

**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.

**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.

**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}

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