21. A "c" is used to indicate radians, just as 66099 is used to indicate degrees. Thus is read " radians,” just as 4° is read "A degrees." When the measure of an angle is expressed as some multiple of, e.g., the unit being the radian, common usage has sanctioned the omission of the "c" after the measure; thus, is 2 π π 2 usually written This practice sometimes confuses the beginner. Yet, with a little care no confusion need arise. We need only remember that is a mere number (Art. 8); that in the expression "the angle" or kindred expressions some unit must be implied, and that by common agreement this unit is the radian, so that is often used for 180°, meaning, then, π radians (Art. 20). This does not exclude the angle ; but if a degree is the unit, it is so expressed. π Another agreement sometimes made, is that the Greek letters a, B, etc., are used to express the measure of angles when the radian is the unit of measure; while, the Roman letters A, B, C, etc., are used when the unit of measure is the degree. The distinction, how ever, is not universally observed. 22. To find the number of degrees in an angle containing a given number of radians. Let we have a = 180 which was to be found. We may in a similar manner find the number of radians in an angle containing a given number of degrees. Or, simply solving (1) for a, D the number of radians in an angle containing D (Art. 17.) degrees. D° «c The problem may be thus solved: 180° T-S the ratio of the same angle to two right angles. (1) for each fraction is EXAMPLE. Find number of degrees in two radians. D 2 180 1. 8° 15' 27". 2. 6° 4' 30". 7. .01375 right angles. 8. .0875 right angles. 9. 1.704535 right angles. 3π 4 3. 1c. 23. EXAMPLES. III.) I. Express each of the following angles as the decimal of a right angle: 3. 97° 5' 15". 5. 132° 6'. 4. 16° 14' 19". 6. 49°. 2. Express in degrees, minutes, and seconds: 1. π II. Express the following angles in rectangular measure: 7. 0. 5. 3.14159265o, etc. 8. .00314159c, 6. 4. (2 n + 1) π. V. Find the ratio of 1. 45° to 3. 11 π 2o 360 .. D = π = 10. .240025 right angles. III. Express the following angles in circular measure: 1. 180°. 4. 2210. 2. 360°. 5. 1°. 3. 60°. 6. 57.295°, etc. IV. Give a geometrical representation of the following trigonometrical angles. Draw the initial line from O, horizontally, to the right. 2. 8. 3. 5π. 2. 1° to 1°. We have arc RP arc RS arc RP the radius 5. (2 n + 1) π. 9. 10π. = 7. n°. 90° 8. π 9. A. etc. 3. 1.75 to 24. Since angles at the centre of a circle are to one another as the arcs on which they stand (Geom.), therefore (see Fig. 6) an angle ROP 100° π Hence the angle ROP= radians. So that the circular measure of an angle (at the centre of a circle) is the ratio of its arc to the radius. 25. L .. the angle: P FIG. 6. 463 EXAMPLE. Find the number of degrees in the angle subtended by an arc 46 ft. 9 in. long, at the centre of a circle whose radius is 25 feet. The angle stands on an are of 463 feet, and the radian, at the centre of the same circle, stands on an arc of 25 feet. RADIAN = 187 100 S 187 180° RADIUS 2 right angles R π 105.8° nearly. EXAMPLES. IV. (IN THE ANSWERS 2 IS USED FOR π.) 1. Find the number of radians in an angle at the centre of a circle of radius 25 feet, which stands on an arc of 37 feet. 2. Find the number of degrees in an angle at the centre of a circle of radius 10 feet, which stands on an arc of 5 feet. 3. Find the number of right angles in the angle at the centre of a circle of radius 3 inches, which stands on an arc of 2 feet. 4. Find the length of the arc subtending an angle of 4 radians at the centre of a circle whose radius is 25 feet. 5. Find the length of an arc of 80 degrees on a circle of 4 feet radius. 6. The angle subtended by the diameter of the sun at the eye of an observer is 32'; find approximately the diameter of the sun if its distance from the observer be 90,000,000 miles. 7. A railway train is travelling on a curve of half a mile radius at the rate of 20 miles an hour; through what angle has it turned in 10 seconds ? 8. A railway train is travelling on a curve of two-thirds of a mile radius, at the rate of 60 miles an hour; through what angle has it turned in a quarter of a minute? 9. Find approximately the number of seconds contained in the angle which subtends an arc one mile in length at the centre of a circle whose radius is 4000 miles. 10. If the radius of a circle be 4000 miles, find the length of an arc which subtends an angle of 1" at the centre of the circle. 11. If in a circle whose radius is 12 ft. 6 in. an arc whose length is .6545 of a foot subtends an angle of 3 degrees, what is the ratio of the diameter of a circle to its circumference? 12. If an arc 1.309 feet long subtend an angle of 7 degrees at the centre of a circle whose radius is 10 feet, find the ratio of the circumference of a circle to its diameter. 13. On a circle 80 feet in radius it was found that an angle of 22° 30' at the centre was subtended by an arc 31 ft. 5 in. in length; hence calculate to four decimal places the numerical value of the ratio of the circumference of a circle to its diameter. 14. If the diameter of the moon subtend an angle of 30', at the eye of an observer, and the diameter of the sun an angle of 32', and if the distance of the sun be 375 times the distance of the moon, find the ratio of the diameter of the sun to that of the moon. 15. Find the number of radians (i.e. the circular measure of) in 10" correct to 3 significant figures. (Use for T.) 16. Find the radius of a globe such that the distance measured upon its surface between two places in the same meridian, whose latitudes differ by 1° 10', may be one inch. 17. By construction prove that the unit of circular measure is less than 60°. 18. On the 31st December the sun subtends an angle of 32′ 36", and on 1st July an angle of 31' 32"; find the ratio of the distances of the sun from the observer on those two days. 19. Show that the measure of the angle at the centre of a circle of radius r, k.a where k depends solely on the unit of angle which stands on an arc a, is employed. Find k when the unit is (i.) a radian, (ii.) a degree. 20. The difference of two angles is, and their sum 56°; find them. 21. Find the number of radians in an angle of n'. 22. Express in right angles and in radians the angles (i.) of a regular hexagon, (ii) of a regular octagon, 23. Taking for unit the angle between the side of a regular quindecagon and the next side produced, find the measures (i.) of a right angle, (ii.) of a racian. 24. Find the unit when the sum of the measures of a degree and of the hundredth part of a right angle is 1. 25. What is the unit when the sum of the measures of a right angles and of b degrees is c? 26. The three angles of a triangle have the same measure when the units are of a right angle, of a right angle, and a radian respectively; find the 0 measure. 27. The interior angles of an irregular polygon are in A.P.; the least angle is 1200; the common difference is 50; find the number of sides. 27. THE TRIGONOMETRICAL RATIOS O I. FIG. 9. CHAPTER III II. P OM 26. Given any angle, XOP, as in the figures. From P, any point in OP, the revolving line, a perpendicular PM is let fall on the initial line OX, as in Figs. 6 and 9, or on the initial line produced, as in Figs. 7 and 8. Six ratios may now be written, viz., MP OP IV., V., VI. are respectively the reciprocals of I., II., and III. III. FIG. 8. MP IV. M FIG. 10. V. 28. These ratios are of much importance in all mathematical investigations. For convenience, they have been given the following names. If XOP be the angle A, then |