(1) is the sine analogy, or sin Ay. (3) is the tangent analogy, or tan Ay. By T1, VA1H1, cos h = cos a cos b, by (3). (4) cos h = cos a cos b = cot A cot B. The first is the Pythagorean analogy. Dividing sin A (1) by cos B (2), using (4). (5) sin A= cos B: cos b, sin B = cos A : cos a. (5) is the complementary analogy. NOTE. If a sphere of unit radius be described about Vas a centre, the faces will cut out a right spherical triangle having the sides a, b, and h. 13 NEGATIVE ANGLES. NOTE. A point moving to the right generates a + distance, but moving back to the left tends to destroy this, and passing the origin generates a distance. A straight line revolving in the order of the quadrants I, II, III, IV, generates a + angle, but revolving back tends to destroy this, and passing the initial line generates a angle. RULE. Changing the sign of an angle changes the sign of its sine, but not of its cosine. .. changes that of its tangent. +< I. II. III. IV. -2 -I. —II. —III. —IV. Since the terminus is changed across the initial line II', but not across the transverse line TT'. That is, I'= IV, 11'=III, III'=II, IV'= I. FUNCTIONS OF nr + A. RULE. If an acute angle be added to or subtracted from an even number of quadrants, the functions of the resulting angle are equal arithmetically in value to the like-named functions of the acute angle; but if an acute angle be added to or subtracted from an odd number of quadrants, the functions of the resulting angle are arithmetically equal in value to the co-named functions of the acute angle. By the equality of the eight possible ratio triangles, and the fact that for an even number of quadrants a and o are the same as for A, but for an odd number they are inter A P 360-A -270 +A NOTE. By revolving 1 and 1– through any number of right angles, one rotation changes sine to cosine, two restores, and so on. CONSTRUCTION. (1) Lay off from the vertex V of a right trihedral a unit on each edge (VH being edge of rt. ≤). (2) Through the extremity of one of the acute edges, as B1, pass a plane to the other acute edge VA, thus: Draw BHVH, then HA VA, lastly join AB. (By Geom.) BAH is the plane measure of dihedral having edge VA. 2 (3) Through the other extremities H, and A,, pass planes || to A1B1H1, ... to VA. (4) By p. 12, the parts of the nine right triangles are as given. NOTE. Napier's Circular Parts are: The two sides about the right angle, the complements of the opposite angles, and the complement of the hypotenuse. His rules are: RULE I. The sine of the middle part is equal to the product of the tangents of the adjacent parts. RULE II. The sine of the middle part is equal to the product of the cosines of the opposite parts. (4) | cos h = cosa cos b = cot A cot B|(4) I. By (Comp. Ay.) An oblique angle and its opposite side are in the same quadrant. II. By (P. Ay.) h<90° when a and b are in the same quadrant. h> 90° when a and b are in different quadrants. 6. The distance of the moon being h, and earth's radius a A= 57' 2"; find h. For the sun, A= 8.8". 7. What is the length of the horizontal shadow of the Washington Monument, when the altitude of the sun is 50°? 8. What is the radius of the circle of latitude on which you live? 9. (a) The angle of elevation of the top of a spire 500 feet distant is measured and found to be 13°. What is its height [above the instrument]? (b) The elevation of base of spire is 9°. What is its height? Spherical Right Triangles. First find a from A and h, then A from a and h. That is, solve in the order that the formulas are given (p. 13). The right ascension R, declination d, and longitude L, of the sun form a right triangle of which these are b, a, and h; A being the obliquity of the ecliptic. 11. L=214° 14' 45"; find R and d. 12. R18 hrs. 44 min. 50 sec.; find L and d. 13. R= 4 hrs. 38 min. 0.88 sec., d= 22° 7' 13.7"; find L and A. REMARKS AND QUESTIONS. Many points rest directly upon page 12. Thus pages 13, 14, 18, 19, and 23 in great part. As the last of 24, and most of 25, require 19 and 20, the given order has been followed. It is very important for the student to observe as to what rests directly on pages 12 and 13. |