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.
Item | Discussion |
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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. |
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