The sides of triangle AMP are proportional to 1, 1, √2. Hence, in order to produce the ratios of 45° quickly, it is merely necessary to draw Fig. 10; from this figure the ratios of 45° can be read off at once. 30° B 30° A A 60 FIG. 11. 60 1 FIG. 12. B. Ratios of 30° and 60°. Let ABC be an equilateral √3 triangle. From any vertex B draw a perpendicular BD to the opposite side AC. Then angle DAB = 60°, angle ABD 30°. = If AB = 2a, then AD = a, and DB = √4a2 — a2 = a√3. By using the same figure it can be shown that In ADB the sides opposite to the angles 30°, 60°, 90°, are respectively proportional to 1, √3, 2. Hence, in order to produce the ratios of 30°, 60°, at a moment's notice, it is merely necessary to draw Fig. 12, from which these ratios can be immediately read off. C. Ratios of 0° and 90°. The algebraical note, Art. 76, may be read now. 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: = 0, it follows that: = 0, tan 0° = 0 a a As MAP approaches 90°, AM approaches zero, and MP approaches to an equality with AP. Therefore, when MAP = 90°, 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 4. 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, cos1x, tan3 x, sec3 x, represent or mean (cos x)3, (tan x)3, (secx)‡. 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 COS X 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 The anti-cosine of x," "the inverse cosine of x,' ""cosine minus one x." number 1, which appears in cos-1x, is not an exponent at all, but is merely part of a symbol. The two ways, (a), (c), of expressing the same idea can be indicated still more briefly by equations, viz., sin A, A = sin-1 §. 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-1 x, cos-1x, ..., are considered in Art. 88. Ex. Express sin A =, cos x = &, tan C = 4, 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. 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 AP2, AM2, MP, 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 the use of tables. Thus, for example, since cos A 1 sec A' a table of natural cosines can be transformed into a table of natural secants by merely taking the reciprocals of the cosines. Again, in logarithmic computation, |