Law of sines, 50; law of cosines, 52; law of tangents, 52. Solu- tion Case I., when one side and two angles are given, 54; Case II., when two sides and the angle opposite to one of them are given, 56; Case III., when two sides and the included angle are given, 60; Case IV., when the three sides are given, 64; area of a triangle, 68; mis- CHAPTER V. THE RIGHT SPHERICAL TRIANGLE: Introduction, 103; formulas relating to right spherical triangles, 105; Napier's rules, 108. Solution: Case I., when the two legs are given, 110; Case II., when the hypotenuse and a leg are given, 110; Case III., when a leg and the opposite angle are given, 111; Case IV., when a leg and an adjacent angle are given, 111; Case V., when the hypotenuse and an oblique angle are given, 111; Case VI., when the two oblique angles are given, 111; solution of the isosceles spherical triangle, 116; solution of a regular spherical polygon, 116. CHAPTER VI. THE OBLIQUE SPHERICAL TRIANGLE: Fundamental formulas, 117; formulas for half angles and sides, 119; Gauss's equations and Napier's analogies, 121. Solution: Case I., when two sides and the included angle are given, 123; Case II., when two angles and the included side are given, 125; Case III., when two sides and an angle opposite to one of them are given, 127; Case IV., when two angles and a side opposite to one of them are given, 129; Case V., when the three sides are given, 130; Case VI., when the three angles are given, 131; area of a spherical triangle, 133. CHAPTER VII. APPLICATIONS OF SPHERICAL TRIGONOMETRY: Problem, to reduce an angle measured in space to the horizon, 136; problem, to find the distance between two places on the earth's sur- face when the latitudes of the places and the difference of their longi- tudes are known, 137; the celestial sphere, 137; spherical co-ordinates, the astronomical triangle, 142; astronomical problems, 143–146. PLANE TRIGONOMETRY. CHAPTER I. TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES. § 1. DEFINITIONS. THE sides and angles of a plane triangle are so related that any three given parts, provided at least one of them is a side, determine the shape and the size of the triangle. Geometry shows how, from three such parts, to construct the triangle and find the values of the unknown parts. Trigonometry shows how to compute the unknown parts of a triangle from the numerical values of the given parts. Geometry shows in a general way that the sides and angles of a triangle are mutually dependent. Trigonometry begins by showing the exact nature of this dependence in the right triangle, and for this purpose employs the ratios of its sides. Let MAN (Fig. 1) be an acute angle. If from any points B, D, F,..... in one of its sides perpendiculars BC, DE, FG,..... are let fall to the other side, then the right triangles ABC, ADE, AFG,..... thus formed have the angle A common, and are therefore mutually equiangular and similar. Hence, the ratios of their corresponding sides, pair by A pair, are equal. That is, M F D B C E G N Fig. 1. Hence, for every value of an acute angle A there are certain numbers that express the values of the ratios of the sides in all right triangles that have this acute angle A. There are altogether six different ratios: I. The ratio of the opposite leg to the hypotenuse is called the Sine of A, and is written sin A. II. The ratio of the adjacent leg to the hypotenuse is called the Cosine of A, and written cos A. III. The ratio of the opposite leg to the adjacent leg is called the Tangent of A, and written tan A. IV. The ratio of the adjacent leg to the opposite leg is called the Cotangent of A, and written cot A. V. The ratio of the hypotenuse to the adjacent leg is called the Secant of A, and written sec A. VI. The ratio of the hypotenuse to the opposite leg is called the Cosecant of A, and written csc A. These six ratios are called the Trigonometric Functions of the C = angle A. EXERCISE I. 1. What are the functions of the other acute angle B of the triangle ABC (Fig. 2)? 2. Prove that if two angles, A and B, are complements of each other (i.e., if A+B=90°), then, 4. What condition must be fulfilled by the lengths of the three lines a, b, c (Fig. 2) in order to make them the sides of a right triangle? Is this condition fulfilled in Example 3? 5. Find the values of the functions of A, if a, b, c respectively have the following values: 6. Prove that the values of a, b, c, in (i.) and (ii.), Example 5, satisfy the condition necessary to make them the sides of a right triangle. 7. What equations of condition must be satisfied by the values of a, b, c, in (iii.) and (iv.), Example 5, in order that the values may represent the sides of a right triangle? Compute the functions of A and B when, 8. a=24, b=143. 9. a 0.264, c=0.265. = 10. b=9.5, c= = 19.3. 11. a=√p2+q, b=√2pq. 12. a= ·√p2+pq, c=p+q· 13. b=2√pq, c=p+q. 28. In a right triangle, c=2.5 miles, sin A=0.6, cos A = 0.8; compute the legs. 29. Construct (with a protractor) the 20°, 40°, and 70°; . determine their functions by measuring the necessary lines, and compare the values obtained in this way with the more correct values given in the following table: 30. Find, by means of the above table, the legs of a right triangle if A = 20°, c = 1; also, if A = 20°, c = 4. = 31. In a right triangle, given a 3 and c=5; find the hypotenuse of a similar triangle in which a 240,000 miles. 32. By dividing the length of a vertical rod by the length of its horizontal shadow, the tangent of the angle of elevation of the sun at the time of observation was found to be 0.82. How high is a tower, if the length of its horizontal shadow at the same time is 174.3 yards? |