5. Express in radians one of the interior angles of a regular octagon; dodecagon. 6. On a circle of 50 ft. radius an arc of 10 ft. is laid off; how many degrees does the arc subtend at the centre? 7. The earth's equatorial radius is approximately 3963 miles. If two points on the equator are 1000 miles apart, what is their difference in longitude? 8. If the difference in longitude of two points on the equator is 1°, what is the distance between them in miles? 9. What is the radius of a circle, if an arc of 1 foot subtends an angle of 1° at the centre ? 10. In how many hours is a point on the equator carried by the earth's rotation through a distance equal to the earth's radius? 11. The minute hand of a clock is 3 ft. long; how far does its extremity move in 25 minutes? [Take π = 22.] 12. A wheel makes 15 revolutions a second; how long does it take to turn through 4 radians? [Take = 22.] § 2. THE TRIGONOMETRIC FUNCTIONS. 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. A Let MAN (Fig. 1) be an acute angle. If from any points pair, are equal. That is, AC AE AG AC AE AG AB AD AF' BC DE FG etc. These ratios, therefore, remain unchanged so long as the angle A remains unchanged. 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 angle A. To these six ratios are often added the two following functions, which also depend only on the angle A: VII. The versed sine of A is 1 — cos A and is written vers A. VIII. The coversed sine of A is 1-sin A and is written covers A. 1. What are the functions of the other acute angle B of the triangle ABC (Fig. 2)? 2. If A+B= 90°, prove sin A=cos B, cos A=sin B, tan Acot B, cot Atan B, sec A=csc B, csc A=sec B, vers A =covers B, covers A=vers B. 3. Find the values of the functions of A, if a, b, c respectively have the following values: (i.) 3, 4, 5. (ii.) 5, 12, 13. (iii.) 8, 15, 17. (v.) 3.9, 8, 8.9. (vi.) 1.19, 1.20, 1.69. 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, 21. Find b if cot A4 and a=17. 22. Find c if sec A=2 and b = 20. Construct a right triangle: given, 24. c=6, tan A=3. 26. 62, sin A=0.6. 25. a=3.5, cos A=1. 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? § 3. REPRESENTATION OF THE FUNCTIONS BY LINES. The functions of an angle, being ratios, are numbers; but we may represent them by lines if we first choose a unit of length, and then construct right triangles, such that the |