DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS 5. The trigonometric functions are numbers, and are defined as the ratios of lines. Let the angle AOP be so placed that the initial line is horizontal, and from P, any point of the terminal line, draw PS perpendicular to the initial line. На SA A To the above may be added the versed sine (written versin) and coversed sine (written coversin), which are defined as follows: - sin x. versin x = 1 − cos x; coversin x =I The values of the sine, cosine, etc., do not depend upon what point of the terminal line is taken as P, but upon the angle. For the triangles OSP and OS'P' being similar, the ratio of any two sides of OS'P' is equal to the ratio of the corresponding sides of OSP. Def. The sine, cosine, tangent, cotangent, secant, and cosecant of an angle are the trigonometric functions of the angle, and depend for their value on the angle alone. 6. A line may by its length and direction represent a number; the magnitude of the number is expressed by the length of the line; the number is positive or negative according to the direction of the line. 7. In 85, if the denominators of the several ratios be taken equal to unity, the trigonometric functions will be rep the line, that is, the ratio of the line to its unit of length. Hence SP may represent the sine of r. In a similar manner the other trigonometric functions may be represented by lines. In the following figures a circle of unit radius is described about the vertex O of the angle AOP, this angle being denoted by x. Then from § 5 it follows that Sin Cos FIG. 2 Sin S Tan O A Tan A SP represents the sine of x. OC represents the cosecant of x. For the sake of brevity, the lines SP, OS, etc., of the preceding figures are often spoken of as the sine, cosine, etc. Hence, we may also define the trigonometric functions in general terms as follows: If a circle of unit radius is described about the vertex of an angle, (1.) The sine of the angle is represented by the perpendicular upon the initial line from the intersection of the terminal line with the circumference. (2.) The cosine of the angle is represented by the segment of the initial line extending from the vertex to the sine. (3.) The tangent of the angle is represented by a line tangent to the circle at the beginning of the first quadrant, and extending from the point of tangency to the terminal line. (4.) The cotangent of the angle is represented by a line tangent to the circle at the beginning of the second quadrant, and extending from the point of tangency to the terminal line. (5.) The secant of the angle is represented by the segment of the terminal line extending from the vertex to the tangent. (6.) The cosecant of the angle is represented by the segment of the terminal line extending from the vertex to the cotangent. The definitions in § 5 are called the ratio definitions of the trigonometric functions, and those in § 7 the line definitions. The introduction of two definitions for the same thing should not embarrass the student. We have shown that they are equivalent. In some cases it is convenient to use the first definition, and in other cases the second, as the student will observe in the course of this study. It is therefore important that he should become familiar with the use of both. SIGNS OF THE TRIGONOMETRIC FUNCTIONS 8. Lines are regarded as positive or negative according to their directions. Thus, in the figures of § 5, OS is posi tive if it extends to the right of O along the initial line, negative if it extends to the left; SP is positive if it extends upward from OA, negative if it extends downward. OP, the terminal line, is always positive. The above determines, from § 5, the signs of the trigonometric functions, since it shows the signs of the two terms of each ratio. By the line definitions the signs may be determined directly. The sine and tangent are positive if measured upward from OA, and negative if measured downward. The cosine and cotangent are positive if measured to the right from OB, and negative if measured to the left. P FIG. 4 FIG. 3 |