Defining Tangents and Cotangents

Table 12.5 provides you with a summary of how to generate tangent and cotangent values. To generate a tangent value, you divide sin θ by cos θ. To generate a secant value, you use the reciprocal of cos θ. A variety of approaches to arriving at the different values of the trigonometric functions exist. The approach given in Table 12.5 proves one of the easiest to follow.

Table 12.5. Generating Tangent and Cotangent Values

Item	Discussion
	If the value of the of cos θ in this function is 0, then the value of the tangent is undefined. When you plot tangent values on a Cartesian plane, the resulting curve rises indefinitely as it approaches a line extending vertically from any point on the x coordinate at which cos θ is 0. To formally state this, you can say that tanu is not defined at any value of + kπ. In this case, k is any integer value. As you see in Figure 12.18, such values are , −, and . Given this situation, the period of the tangent values is π.
	If the value of the of sin θ is 0, then the value of the cosecant is undefined. When you plot cotangent values on a Cartesian plane, the resulting curve rises indefinitely as it approaches a line extending vertically from any point on the x coordinate at which sin θ is 0. A formal way to say this is that csc θ is not defined at any value of π + kπ. The value of k is any integer. As you see in Figure 12.19, such values are π, 2π, −π, −2π, 3π, and 0. Given this situation, the period of the tangent values is π.
	The cosine and secant functions are reciprocals of each other. The value of the cosecant is undefined when it falls on a vertical line passing through a point on the x axis at which the cosine value is 0. Given this situation, the period of the secant values is 2π.
	The sine and cosecant functions are reciprocals of each other. The value of the cosecant is undefined when it falls on a vertical line passing through a point on the x axis at which the sine value is 0. Given this situation, the period of the cosecant values is 2π.