MEASUREMENT OF ANGLES, SEXAGESIMAL, CENTESIMAL, 1. IN geometry angles are measured in terms of a 2. In the Sexagesimal system of measurement a right angle is divided into 90 equal parts called Degrees. Each degree is divided into 60 equal parts called Minutes, and each minute into 60 equal parts called The symbols 1°, 1′, and 1′′ are used to denote a degree, This system is well established and is always used in the practical applications of Trigonometry. It is not 1 3. On this account another system of measurement called the Centesimal, or French, system has been proposed. In this system the right angle is divided into 100 equal parts, called Grades; each grade is subdivided into 100 Minutes, and each minute into 100 Seconds. The symbols 18, 1', and 1" are used to denote a Grade, a Minute, and a Second respectively. Thus 100 Seconds (100") make One Minute (1`), - 4. This system would be much more convenient to use than the ordinary Sexagesimal System. As a preliminary, however, to its practical adoption, a large number of tables would have to be recalculated. For this reason the system has in practice never been used. - 5. To convert Sexagesimal into Centesimal Measure, and vice versa. Since a right angle is equal to 90° and also to 100%, we have Hence, to change degrees into grades, add on oneninth; to change grades into degrees, subtract one-tenth. If the angle do not contain an integral number of degrees, we may reduce it to a fraction of a degree and then change to grades. In practice it is generally found more convenient to reduce any angle to a fraction of a right angle. The method will be seen in the following examples; Ex. 1. Reduce 63° 14' 51" to Centesimal Measure. Ex. 2. Reduce 94o 23' 87" to Sexagesimal Measure. 94% 23' 87``='942387 right angle 90 84.81483 degrees 60 48.8898 minutes 60 53.3880 seconds .. 948 23' 87" 84° 48′ 53.388". 6. Angles of any size. Suppose AOA' and BOB' to be two fixed lines meeting at right angles in O, and suppose a revolving line OP (turning about a fixed point at 0) to start from OA and revolve in a direction opposite to that of the hands of a watch. For any position of the revolving line between OA and OB, such as OP1, it will have turned A P3 раз P2 B P1 B' P A P4 through an angle AOP1, which is less than a right angle. For any position between OB and OA', such as OP2, the angle AOP, through which it has turned is greater than a right angle. For any position OP, between OA' and OB', the angle traced out is AOP, i.e. AOB+BOA' + A'OP3, i.e. 2 right angles + A'OP, so that the angle described is greater than two right angles. For any position OP, between OB' and OA, the angle turned through is similarly greater than three right angles. When the revolving line has made a complete revolution, so that it coincides once more with OA, the angle through which it has turned is 4 right angles. If the line OP still continue to revolve, the angle through which it has turned, when it is for the second time in the position OP1, is not AOP, but 4 right angles +AOP1. Similarly when the revolving line, having made two complete revolutions, is once more in the position OP2, the angle it has traced out is 8 right angles + AÓP 2. 7. If the revolving line OP be between OA and OB it is said to be in the first quadrant; if it be between OB and OA' it is in the second quadrant; if between OA′ and OB′ it is in the third quadrant; if it is between OB′ and OA it is in the fourth quadrant. 8. EX. What is the position of the revolving line when it has turned through (1) 225°, (2) 480°, and (3) 1050°? (1) Since 225°=180°+45°, the revolving line has turned through 45° more than two right angles and is therefore halfway between OA' and OB'. (2) Since 480°= 360°+120°, the revolving line has turned through 120° more than one complete revolution, and is therefore between OB and OA', and makes an angle of 30° with OB. (3) Since 1050° = 11 × 90° +60°, the revolving line has turned through 60° more than eleven right angles and is therefore between OB' and OA and makes 60° with OB'. Express in grades, minutes, and seconds the angles Express in terms of right angles and also in degrees, minutes, and of the revolving line when it has traced out the 19. 31 right angles. 20. 13 right angles. 23. 745°. 24. 1185°. 25. 150o, 21. 120°. 22. 315°. 26. 4208. 27. 875%. 28. How many degrees, minutes and seconds are respectively passed over in 11 minutes by the hour and minute hands of a watch? 29. The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both the angles in degrees. 30. Prove that the number of Sexagesimal minutes in any angle is to the number of Centesimal minutes in the same angle as 27 : 50. 31. Divide 44° 8' into two parts such that the number of Sexagesimal seconds in one part may be equal to the number of Centesimal seconds in the other part. Circular Measure. 9. A third system of measurement of angles has been devised, and it is this system which is used in all the higher branches of Mathematics. |