15.] TRIGONOMETRIC RATIOS OF 0° AND 90°. 31 Let the hypotenuse in each of the right-angled triangles in Fig. 13 be equal to a. A M It is apparent from this figure that if the angle MAP approaches zero, then the perpendicular MP approaches zero, and the hypotenuse AP approaches to an equality with AM; so that, finally, if MAP = 0, then MP = 0, and AP = AM. Therefore, when MAP = FIG. 13. 0, it follows that: As MAP approaches 90°, AM approaches zero, and MP approaches to an equality with AP. Therefore, when MAP 90°, it follows that: = N.B. Read the first few lines of Art. 17 before attacking the problems. 8. 2 sin5 30° tan3 60° cos3 0°. 7. 10 cost 45° sec 30°. 9. x cots 45° sec2 60° = 11 sin2 90°; find x. 10. x(cos 30° + 2 sin 90° + 3 cos 45° - sin2 60°) = 2 sec 0° - 5 sin 90°; find x. 16. Relations between the trigonometric ratios of an acute angle and those of its complement. When two angles added together make a right angle, the two angles are said to be complementary, and each angle is called the complement of the other. For example, the acute angles in a right-angled triangle are complementary; the complement of A is 90° – A; the complement of 27° is 63°. Ex. 1. What are the complements of 10°, 12° 30', 47°, 56° 27', 35° ? In Fig. 2, Art. 12, the angle APM is the complement of the angle A. Now, Comparison of these ratios with the ratios of A in (2), Art. 12, These six relations can be expressed briefly: Each trigonometric ratio of an angle is equal to the corresponding co-ratio of its complement. Ex. Compare the ratios of 30° and 60°; of 0° and 90°. 17. Exponents in trigonometry. When a trigonometric ratio has an exponent, a particular way of placing the exponent has been adopted. For example, (sin x)2 is written sin2 x. There is no ambiguity in the second form, and the advantage is apparent. Thus, cos x, tan3 x, secx, represent or mean (cos x)(tan x)3, (sec x). There is one exponent, however, which must not be written with the brackets removed. This exception is the exponent 1. Thus, for example, (cos x)-1, which means must never be written cos-1 x. The reason 1 for this is that the symbol cos-1x is used to represent something else. This symbol denotes the angle whose cosine is x, and is read thus, or is read "the 16-18.] RELATIONS BETWEEN RATIOS. 33 The anti-cosine of x,' ""the inverse cosine of x, ""cosine minus one x." number - 1, which appears in cos-1 x, is not an exponent at all, but is merely part of a symbol. Suppose that (a) "the sine of the angle A is .' The latter idea can also be expressed by saying (b) "A is the angle whose sine is "; or, more briefly, by saying, (c) "A is the anti-sine of §." The two ways, (a), (c), of expressing the same idea can be indicated still more briefly by equations, viz., sin A, A = sin-13. Thus, (sin x)-1 and sin-1x mean very different things; for (sin x)-1 is 1 which is a number, and sin-1 x is an angle. sin x NOTE. The symbols sin-1x, cos-1x, ..., are considered in Art. 88. Ex. Express sin A, cos x = , tan C4, sec A = 9, cosec A = 17, in the inverse form. 18. Relations between the trigonometric ratios of an acute angle. [N.B. Some relations between these ratios may have been noticed or discovered by the student in the course of his preceding work. If so, they should now be collected, so that they can be compared with the relations shown in this article.] Some of the preceding exercises have shown that when one trigonometric ratio of an angle is known, the remaining five ratios can be easily determined. This at least suggests that the ratios are related to one another. In what follows, A denotes any acute angle. A. Reciprocal relations between the ratios. Inspection of the definitions (3), Art. 12, shows that: B. The tangent and cotangent in terms of the sine and cosine. In the triangle AMP (Fig. 2, Art. 12), C. Relations between the squares of certain ratios. In the triangle AMP (Fig. 2, Art. 12), indicating by MP2 the square of the length of MP, MP2 + AM2 = AP2. On dividing each member of this equation by AP, AM3, MP2, in turn, there is obtained In reference to the angle A, these equations can be written: NOTE 1. The relations shown above are true, not only for acute angles, but for all angles. This is shown in Art. 44. NOTE 2. Relations (1) have a practical bearing on the construction and 1 the use of tables. Thus, for example, since cos A = a table of natural sec A' cosines can be transformed into a table of natural secants by merely taking the reciprocals of the cosines. Again, in logarithmic computation, 18.] EXAMPLES. 35 NOTE 3. An equa ion involving trigonometric ratios is a trigonometric equation. Thus, for example, tan A = 1. One angle which satisfies this equation is the acute angle A = 45°. Other solutions can be found after Arts. 84-87 have been taken up. EXAMPLES. A few simple exercises are given below, the solution of which brings in the relations shown in this article. These exercises are algebraic in character; collections of exercises of this kind are given also in other places in this book. In the following examples, the positive values of the radicals are to be taken. The meaning of the negative values is shown in Art. 44. 1. Given that sin A = 1, find the other trigonometric ratios of A by means of the relations shown in this article. These results may be verified by the method used in solving Exs. 1, 5–7, Art. 14. 2. Express all the ratios of angle A in terms of sin A. sec1 A − 1 = (sec2 A)2 − 1 = (1 + tan2 A)2 − 1 = 2 tan2 A + tan1 A. 3 cosece= : 0. = 2 sec2 A. 2 = = 2 sec2 A. cos2 A |