77. Functions of Acute Angles. A special set of definitions for the functions of acute angles, which are sometimes useful and should be known, follows directly as a special

case of the general definitions given above. Thus, Fig. 4, in which the angle XOP lies in a right triangle,

These definitions, it must be noted, completely agree with the more general definitions, but are applicable only to angles less than ninety degrees, since angles greater than ninety degrees cannot occur in right triangles.

8. Reciprocal Functions. Two questions would naturally suggest themselves at this point: Are the trigonometric

way? and, second, if there be a definite relation between two given angles will the functions of those angles bear some special relation to each other? We shall proceed to answer the first of these questions affirmatively, but shall leave the discussion of the second question to a later chapter (Chap. II). Thus, if a be any angle, it follows by the definitions of the trigonometric functions that

or, the sine and cosecant, the cosine and secant, the tangent and cotangent respectively of the same angle are reciprocals of each other.

9. Tangent, Sine and Cosine. Again, by definition, and by

These relations may be proved otherwise, thus, Fig. 5:

10. Sine and Cosine. Also, Fig. 5, it is obvious that

(sin XOP)2+(cos XOP)2 = 1

or as it is usually written, letting α = Z XOP,

sin2 a + cos2 α = 1.

11. Tangent and Secant. Similarly, writing the first equation of Art. 10 in the form

OP2 = MP2 + OM2

and dividing each term by OM2, we have

OP\2

MP\2

+1.

12. Fundamental Relations. These relations, summarized below, are of great importance and must be memorized.

13. By means of the identities of Art. 12 the value of any one of the trigonometric functions may be expressed in terms of each of the other five. Thus, by (3)

where the radical may be either plus or minus.

14. To Compute the Values of the Other Functions when One Function of an Angle is Given. By means of the relations of the preceding article if the value of any one function of an angle be given, the values of the remaining functions may be found, but a simpler method of obtaining them is

Example 1. Given sin A, find the values of the remaining functions.

The distance being always positive, the minus sign necessarily is taken with the ordinate.

Therefore, Fig. 6,

This is

tance is 3, or any multiple (m) of — 2 and 3. The third side of the right triangle is ±√9-4=±√5. the value of the abscissa and we may write the values of the six functions from the definitions.