French measure, and conversely. Let E and F be the numbers of degrees and grades respectively in the same angle— subdivisions of a degree and of a grade being expressed as decimals. Then, since there are 90° and 100o in a right angle, EF: 90: 100, Ex. 1. Convert 34° 20′ 24′′ into French measure. 34° 20′ 24′′ 34°•34; hence the number of grades in the angle = 10 × 34.34=38.1555...... therefore, in French measure, the angle is 38o 15' 55"5. Ex. 2. Convert 559 18' 35" into English measure. hence the angle is, in English measure, 49° 39′ 54′′·54. EXAMPLES III. 1. Express the angle of a regular octagon in (1) degree measure, (2) radian measure in terms of π, (3) French measure. 2. A ship is steering N.N.W., and the course is changed to N.E. Express the change in direction in (1) points, (2) degree measure, (3) radian measure in terms of π, (4) French measure. 3. Show that the number of degrees in the angle of a 4. Show that the number of degrees in the angle of a regular polygon of 342 sides is equal to the number of grades in the angle of a regular polygon of 19 sides. 5. Find, in degrees, the angles of an isosceles triangle in which each of the two equal angles is 4 times the third angle. EXAMPLES A. 1. A, B, C are three lighthouses. From A the bearings of B and C are respectively w.s.w. and s.s.E. If ABAC, show that the bearing of C from B is E.S.E. 2. If n is the measure of an angle when m° is the unit of angle, show that m is the measure of the same angle when no is the unit of angle. 3. If the unit of angle were, what number would represent the magnitude of an angle which is radian? 4. A railway curve AB is an arc of a circle whose radius is 11⁄2 miles and the length of AB is 220 yards; find, in degree measure, the angle ATB between AT and BT, the tangents to the curve at A and B. 5. A ship sails 380 miles due south; find the change in the latitude, taking the earth as a sphere of 4000 miles radius. 6. Two straight lines include an angle, and with their point of intersection as centre concentric circles are described. Show that the lengths of the arcs of these circles intercepted between the two straight lines are proportional to the radii. 7. Two circles touch internally at the point A, the radius of the larger circle being equal to the diameter of the smaller circle; from the centre O of the smaller circle a line is drawn perpendicular to OA, cutting the smaller circle in B and the larger circle in C. Show that the length of the arc AC is of the length of the arc AB. 8. If the unit of angle is n°, what number represents the angle whose radian measure is 0? =10, 0·7854, show that the number is 45, very If n = nearly. 9. Assuming that the moon revolves round the earth in 27 days in a circle whose radius is 240,000 miles, find the (254) В number of feet the moon passes over in one second of time. 10. Being given that the three periods, the anomalistic year, the sidereal year, and the tropical year are in the proportion anomalistic year: sidereal year : tropical year :: 360° 0′ 11′′ 48: 360° : 359° 59′ 9′′ 76, and that the duration of the tropical year is 365 days, 5 hours, 48 minutes, 46 seconds, find the duration of (1) the anomalistic year, (2) the sidereal year. 11. If the angle of a regular polygon of m sides is equal to of the angle of a regular polygon of n sides, show that 12. In the preceding Example, show that m must be one of the numbers 5, 10, 14, 20, 30, 50, 110. Hence show that there are only 7 pairs of polygons which satisfy the conditions of the problem. 13. If the number of degrees in the angle of a regular polygon of m sides is equal to the number of grades in the angle of a regular polygon of n sides, show that n = =20 360 m+18 14. Show that there are 11 pairs of regular polygons which satisfy the conditions of the problem in the preceding Example. CHAPTER II. TRIGONOMETRICAL RATIOS OF ANGLES LESS THAN A RIGHT ANGLE. 12. We are now about to define, with reference to angles, certain ratios which are called trigonometrical ratios. These ratios form the basis of Trigonometry, and it is by means of them that we can combine in equations the angles and sides of plane rectilineal figures. 13. Definitions of Trigonometrical Ratios. 4 B Let AOB be any angle less than a right angle, and let any point P be taken in OA or OB. Taking P in OB, let PM be drawn perpendicular to OA. Then six of the trigonometrical ratios of the angle AOB are defined as the ratios of the lengths of the sides of the right-angled triangle POM. M The sine of the angle AOB is the ratio of the length of PM the cosine of the angle AOB is the ratio of the length of OM the tangent of the angle AOB is the ratio of the length of the cotangent of the angle AOB is the ratio of the length of the secant of the angle AOB is the ratio of the length of the cosecant of the angle AOB is the ratio of the length of The words sine, cosine, tangent, cotangent, secant, and cosecant are usually abbreviated into sin, cos, tan, cot, sec, and cosec. Denoting the angle AOB by the letter A, and, in the triangle POM, calling PM the perpendicular, OM the base, and OP the hypotenuse, the definitions of the above six trigonometrical ratios of the angle AOB are A seventh ratio, the versed sine of the angle A, is defined to be the excess of unity over the cosine of A. The term, versed sine of the angle A, is usually abbreviated into versA, so that the definition is equivalent to the equation versA = 1-cos A. 14. Trigonometrical Ratios depend only on the Angle. B 5 N It is evident from the definitions that the trigonometrical ratios of angles are not lengths but numbers, being the ratios of two lengths; and it will now be shown that these numbers are altogether independent of the position of the point P, and depend only on A the magnitude of the angle AOB. 我 M M' For take another point P' anywhere in OB, and a third point Q in OA; draw P'M' perpendicular to OA, and QN perpendicular to OB. Then the three triangles OPM, OP’M', and OQN are similar triangles, for they are right-angled and have the angle AOB common. Hence the ratio of any two sides in one of the triangles is equal to the ratio of the corresponding sides in each of the other two triangles. For |