高校数学教材についての考察VII : 三角比・三角関数の正接または直線x=kの立場からの考察

抄録

In learning of trigonometric ratios in high school mathematics "Mathematics I", many text books define at first a tangent, next a sine and cosine. A reason to define in this order would come from an educational care since students are familiar with a notion of tangent as a slope, for example, in their daily life. However, in the process of the study, the sine and cosine play a central role in trigonometric ratios and functions. For example, the Sine Theorem and the Cosine Theorem are the important topics in "Mathematics I", and in fact the tangent is simply expressed as the ratio of sine to cosine. Then, we can propose a problem: To consider trigonometric ratios and functions from a viewpoint of tangent. We study this problem by using an intersection of a line through the origin and a line x=k in the orthogonal coordinate plane. We show Tangent Addition Theorem without using Sine and Cosine Addition Theorems. A derivative of tangent function tanx is obtained as 1+ tan^2x by using the Tangent Addition Theorem. A geometric interpretation is given for the property: an integration of a rational function of the sine and cosine functions of x is reduced to an integration of rational function of t by a substitution tan(x/2)=t. As a result, we see that some properties of trigonometric ratios and functions are naturally obtained from a viewpoint of tangent or line x=k.