Trigonometric Functions Definition

Chapter 1.2

The three ratios b/c, a/c, b/a, and others to be studied later, are referred to as the trigonometric functions of an angle. In elementary mathematics a function may be defined as an expression in which the value of one quantity depends upon the value of another. For example, the circumference of a circle is a function of the radius since its value depends upon the length of its radius. The number of books you can buy for ten dollars depends upon the cost of the individual books. Can you discover other illustrations of functional relationships in everyday life or in the sciences?

It is easily seen that the value of the sine of an angle depends upon the size of the angle. Consequently, the sine of an angle is a function of the angle. In a similar manner, the cosine and the tangent of an angle are functions. Of what?

Referring back to the right triangle, let us assume that b = 5 inches and that a = 12 inches. Then AB can be found by the Pythagorean Theorem which you have already studied in geometry:

The square of the hypotenuse of a right triangle is equal to the square of one leg plus the square of the other leg.

Corollary

The square of either leg is equal to the square of the hypotenuse diminished by the square of the other leg.

Hence c2 = 52 + 122 = 169. Therefore c = 13 because the square root of 169 equals 13.

By the definitions given above, cos A = 5/13 or 0.38 * in decimal form. Find in decimal form the values of sin A and tan A. Also find sin B, cos B and tan B. What discovery have you made?

Summary of Definitions of Trigonometric Ratios

Sine

                   opposite leg
sine of an angle = ------------
                   hypotenuse    

Cosine

                      adjacent leg
cosine of an angle =  ------------
                      hypotenuse

Tangent

                       opposite leg
tangent of an angle =  ------------
                       adjacent leg

NOTE - * The degree of precision and accuracy in measurements and the corresponding significant figures to be used in computation are discussed later in this chapter.

Pages 5-6

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