a + b 15. If a, b, c are in continued proportion, show that b + c b, bc are also in continued proportion. 16. If a : 6::c:d, then Jan + 72" : " + ď" :: (a - b)* : (0 - d)". 17. From a vessel containing a cubic inches of hydrogen gas, b cubic inches are withdrawn, the vessel being filled up with oxygen at the same pressure. Show that if this operation be repeated n times successively, the quantity of hydro (a - b)" gen remaining in the vessel is cubic inches, when re a”duced to the original pressure. 18. If, in Ex. 34, page 225, (az, Az, az), (61, 62, 63), and (C1, C2, C3) are corresponding terms respectively, show that babz (cz - Cy) + a2b3b1 (cz - 6) + a,b,ba (- ca) = 0. SEOTION III. PLANE TRIGONOMETRY, CHAPTER I. MODES OF MEASURING ANGLES BY DEGREES AND GRADES. 1. We are able to determine geometrically a right angle, and it might therefore be taken as the unit of angular measurement. Practically, however, it is too large, and so we take a determinate part of a right angle as a standard. In England we divide a right angle into 90 equal parts, called degrees, and we further subdivide a degree into 60 equal parts, called minutes, and again a minute into 60 equal parts, called seconds. This is the English or sexagesimal method. In France the right angle is divided into 100 equal parts, called grades, a grade into a hundred equal parts, called minutes, and a minute into 100 equal parts, called seconds. This is the French or centesimal method, and its advantages are those of the metric system generally. The symbols , ! ", are used to express English degrees, minutes, seconds respectively, and the symbols', )", to express French grades, minutes, seconds respectively. Conversion of English and French Units. 2. Let D the number of degrees in an angle, and G the number of grades in the same angle; D then expresses the angle in terms of a right angle; G and so also does 100 90 Hence the following rules :- From the number of grades SUBTRACT 1), and the remainder is the number of degrees. 2. To convert degrees into grades. To the number of degrees ADD }, and the sum is the number of grades. Ex. 1. Convert 139 18' 75' into English measure. No. of grades 13.1875 Subtract of this 1.31875 :. No of degrees = 11.86875 60 521•12500 = 60 71-500 Ans. 11° 52' 711.5. Ex. 2. Convert 18° 7' 30'' into French measure. No. of degrees 18.125 Add } of this 2.0138 :: No. of grades 20.1388 Ans. 20° 13' 88".g. 3. An angle may be conceived to be generated by the revolution of a line about a fixed point. Thus Let OA be an initial line, and let a line, OP, starting from OA, revolve with O as centre, and take up successively the positions OP, OP, OP, OP A Now the magnitude of an angle may be measured by the amount of turning required to generate it. When, therefore, the revolving line reaches the position OB, we may conceive an angle to have been generated whose magnitude P is two right angles. And, further, when the revolving line assumes the positions BH OP, OP, the angles AOP3, AOP (the letters being read in the direction of revolution) are angles whose magnitudes are each greater than two right angles. Indeed, when the revolving line again reaches the position OP, we may conceive an angle to have been generated whose magnitude is four right angles. Lastly, if the revolution of the line OP be continued, we may conceive of angles being generated to whose magnitude there is no limit. Ex. I. 1. Express 39° 22' 30" in French measure, and 13' 15' 75" in English measure. 2. One of the angles at the base of an isosceles triangle is 50°. Express the vertical angle in grades. 3. Divide an angle of n degrees into two such parts that the number of degrees in one part may be twice the number of grades in the other. 4. Two angles of a triangle are respectively a°, bo, express the other angle in degrees and grades. 5. If of a right angle be the unit of measurement, express an angle which contains 22.5 degrees. 6. Show how to reduce English seconds to French seconds. 7. If the unit of measurement be 8°, what is the value of 10%. 8. If two of the angles of a triangle be expressed in grades, and the third in degrees, they are respectively as the numbers 5, 15, 18. Find the angles. 9. What is the value in degrees and grades of an angle which is the result of the revolution of a line 3} times round. 10. In what quadrants are the following angles found :145°, 96', 327°, 272, 272°, 11. If ao be taken as the unit of angular measurement, express an angle containing b. 12. What is the unit of measurement when a expresses P 9 of a right angle? CHAPTER II. THE GONIOMETRIC FUNCTIONS. 4. It was formerly usual in works on Trigonometry to give the following definitions : Let a circle be described from centre A, I with radius AB supposed to be unity, then (1.) The sine of an arc BC is the perpendicular from one extremity, C, of the arc upon the diameter passing through the other extremity B. Thus CS is the SINE of the arc BC. (2.) The cosine of an arc is the sine of the complement of the arc. Thus, since DC is the complement of BC, S'C is the COSINE of the arc BC. (3.) The tangent of an arc BC is a line drawn from one extremity, B, of the arc touching the circle, and terminated in the diameter which passes through the other extremity, C, of the arc. Thus, BT is the TANGENT of the arc BC. (4.) The cotangent of an arc is the tangent of the complement of the arc. Thus, DT' is the COTANGENT of the arc BC. (5.) The secant of an arc BC is a line drawn from the centre through one extremity, C, of the arc, and terminated in the tangent at the other extremity: Thus, AT is the SECANT of the arc BC. (6.) The cosecant of an arc is the secant of the complement of the arc. Thus, AT'is the COSECANT of the arc BC. |