PREFACE. I HAVE endeavoured in this Elementary Text-book to lead the student by easy gradations from the Measurement of Angles to the Solution of Triangles. Chapter I. is the result of an attempt to treat the subject of Measurement of Angles in a systematic manner. In this chapter I have substituted the terms radian and radian measure, which are now coming into use, for the terms unit of circular measure and circular measure respectively. The work contains a large number of Examples, which have been selected, for the most part, from papers set in the Science and Art, the Civil Service, and the British University Examinations. Examples involving numerical calculations have been freely introduced. With reference to approximate numerical calculations, it may be well to remind the student of the usual rule, which has been followed throughout the work:-Increase by unity the figure in the last decimal place to which the calculation is to be carried, if the next decimal figure would be 5 or greater than 5. The sets of Examples marked A, B, C, &c., and also the Miscellaneous Examples at the end, may be omitted on a first reading. It is hoped that this Text-book will be found useful to students preparing for University Pass Examinations, and other Examinations in which Elementary Trigonometry is included. Any suggestions or corrections will be gratefully received. HOGGANFIELD, GLASGOW, 2d August, 1884. R. H. PINKERTON. CONTENTS. СНАР. 1. Measurement of Angles, Page 5 II. Trigonometrical Ratios of Angles less than a Right Angle, 18 III. Applications of Trigonometrical Ratios, ... 42 IV. Angles of any Magnitude. Ratios of an Angle of any སྐྱ V. Section I. Ratios of the Sum or Difference of Two Angles, 73 Section II. Ratios of Multiples and Sub-Multiples of an VI. Logarithms and Logarithmic Tables, VII. Solution of Right-angled Triangles-Applications, VIII. Section I. General Properties of Triangles, Section II. Solution of Oblique-angled Triangles. IX. Areas. Miscellaneous Propositions, ANSWERS TO THE EXAMPLES, TRIGONOMETRY. CHAPTER I. MEASUREMENT OF ANGLES. 1. Plane Trigonometry is that branch of Mathematics which treats of angles and of the relations between the angles and sides of plane rectilineal figures. 2. Units of Angle. In measuring any magnitude, we have first to choose some definite magnitude of the same kind as our standard or unit of measurement; then the number of times the magnitude we are measuring contains the unit is the numerical measure of the magnitude. It is evident that the same magnitude will be represented by different numbers when different units are adopted. For example, the distance of a mile will be represented by the number 1 when a mile is the unit of length, by the number 1760 when a yard is the unit of length, by the number 5280 when a foot is the unit of length, and so on. In like manner, the number expressing the magnitude of au angle will depend on the unit of angle. Two units of angle are in general use: the degree and the radian. The degree is the unit of angle adopted in actual observations of angles, and generally in practical applications of trigonometry; while the radian is the unit of angle in common use in the theoretical parts of mathematics. 3. Practical Unit of Angle-the Degree. The degree is the angle which is the 90th part of a right angle, and is subdivided into minutes and seconds. The minute of angle is the 60th part of a degree, and the second of angle is the 60th part of a minute of angle. One degree, one minute, and one second are written respectively 1o, 1′, 1′′. An angle which is less than a second is written as the decimal of a second. Thus, the angle which contains 37 degrees, 43 minutes, 25.3 seconds is usually written 37° 43′ 25′′·3. It follows immediately from these definitions that 1 right angle 90°, 2 right angles 180°, an angle of an equilateral triangle, (being of 2 right angles) = 60°, right angle = 45°, An angle which is less than a degree, instead of being expressed in minutes and seconds, may, of course, be written as the fraction of a degree. For example, the angle which is the 90 48 48th part of a right angle contains or 17 degrees, and may be written in any one of the three forms 1°, 1°.875, 1° 52′ 30′′. The student will notice that this division of a degree into minutes and seconds of angle corresponds precisely to the division of an hour into minutes and seconds of time. Ex. 1. Find the number of degrees in each of the angles of a regular pentagon. By Euclid I. 32, Cor. 1, the five angles of a regular pentagon are together equal to (5 × 2−4) or 6 right angles. Hence each Ex. 2. The angles of a triangle are as 1:2:3; find the number of degrees in each. By Euclid I. 32, the three angles of a triangle are together equal to 2 right angles, and 2 right angles = 180°. Hence, since 1+2+3=6, the least angle is, the greatest, and the third of 180°. Therefore the angles are 30°, 60°, 90°. Ex. 3. Two of the angles of a triangle are 23° 47′ 53′′·5 and 102° 39′ 57′′-3; find the third angle. Since the three angles of a triangle are together equal to 2 right angles, we subtract from 180° the sum of the two given angles, or 126° 27′ 50′′8, and the remainder, or 53° 32′ 9′′-2, is the third angle. 23° 47' 53"-5 102° 39' 57":3 180° 126° 27′ 50′′.8 53° 32′ 9"-2 2. In a triangle one angle is a right angle; find the two acute angles in each of the following cases: (1) When one is 5 times the other. (2) When the difference between them is 10°. (3) When they are in the ratio 11: 19. 3. Given, in each of the following cases, two angles of a triangle, find the third angle: (1) 60°, 70°. (2) 40° 50′, 75° 40'. (3) 35° 57′ 59′′, 83° 49′ 23′′. (4) 63° 29′ 56′′ 35, 93° 33′ 25′′ 78. 4. Find the number of degrees, minutes, and seconds in the angle of a regular polygon of 64 sides. 5. The four angles of a quadrilateral are as 12 : 15 : 18 : 19; express them in degrees, minutes, and seconds. 6. Find the angle between the hands of a watch at 20 minutes past 1 o'clock. 4. Ratio of Circumference of a Circle to its Diameter. In all circles the length of the circumference contains the length of the diameter the same number of times. The truth of this we shall assume at present, and return to the proof in a later chapter. Hence, if we divide the number expressing the length of the circumference by the number expressing the length of the diameter, the quotient will be a number which is circumference the same for all circles, or more briefly, the ratio diameter |