70. To prove that the radian measure of any angle at the centre of a circle is expressed by the fraction Let AOC be any angle at the centre of a circle, and AOB a radian; then radian measure of AOC LAOC subtending arc radius 71. If a be the length of the arc which subtends an angle of radians at the centre of a circle of radius r, we have seen in the preceding article that arc radius The fraction is usually called the circular measure of the angle at the centre of the circle subtended by the arc. The circular measure of an angle is therefore equal to its radian measure, each denoting the number of radians contained in the angle. We have preferred to use the term radian measure exclusively, in order to keep prominently in view the unit of measurement, namely the radian. NOTE. The term circular measure is a survival from the times when Mathematicians spoke of the trigonometrical functions of the arc. [See page 80.] Example 1. Find the angle subtended by an arc of 7.5 feet at the centre of a circle whose radius is 5 yards. Let the angle contain 0 radians; then Example 2. In running a race at a uniform speed on a circular course, a man in each minute traverses an arc of a circle which subtends 29 radians at the centre of the course. If each lap is 792 yards, how long does he take to run a mile? 227 T= 7 Let a yards be the length of the arc traversed in each minute; then from the formula a=r0, that is, the man runs 360 yds. in each minute. 1760 44 .. the time= 360 or minutes. Thus the time is 4 min. 53 sec. Example 3. Find the radius of a globe such that the distance measured along its surface between two places on the same meridian whose latitudes differ by 1° 10' may be 1 inch, reckoning that Let the adjoining figure represent a section of the globe through the meridian on which the two places P and Q lie. Let O be the centre, and denote the radius by r inches. 22 = 77 7 EXAMPLES. VII. c. 1. Find the radian measure of the angle subtended by an arc of 16 yards at the centre of a circle whose radius is 24 feet. 2. An angle whose circular measure is 73 subtends at the centre of a circle an arc of 219 feet; find the radius of the circle. 3. An angle at the centre of a circle whose radius is 2.5 yards is subtended by an arc of 7.5 feet; what is the angle? 4. What is the length of the arc which subtends an angle of 1625 radians at the centre of a circle whose radius is 3'6 yards? 5. An arc of 17 yds. 1 ft. 3 in. subtends at the centre of a circle an angle of 19 radians; find the radius of the circle in inches. 6. The flywheel of an engine makes 35 revolutions in a second; how long will it take to turn through 5 radians? = 22 7. The large hand of a clock is 2 ft. 4 in. long; how many inches does its extremity move in 20 minutes? 22 [= 27]. 8. A horse is tethered to a stake; how long must the rope be in order that, when the horse has moved through 52.36 yards at the extremity of the rope, the angle traced out by the rope may be 75 degrees? 9. Find the length of an arc which subtends 1 minute at the centre of the earth, supposed to be a sphere of diameter 7920 miles. 10. Find the number of seconds in the angle subtended at the centre of a circle of radius 1 mile by an arc 5 inches long. 22 11. Two places on the same meridian are 145.2 miles apart; find their difference in latitude, taking π= and the earth's diameter as 7920 miles. י 7 12. Find the radius of a globe such that the distance measured along its surface between two places on the same meridian whose latitudes differ by 13° may be 1 foot, taking = 22 MISCELLANEOUS EXAMPLES. B. 1. Express in degrees the angle whose circular measure is •15708. 2. If C=90°, A=30°, c=110, find b to two decimal places. 3. Find the number of degrees in the unit angle when the 12π angle is represented by 13. 25 4. What is the radius of the circle in which an arc of 1 inch subtends an angle of l' at the centre? 5. Prove that (1) (sin a+cos a) (tan a+cot a) = sec a+cosec a ; (2) (√3+1)(3 – cot 30°) = tan3 60° – 2 sin 60°. 6. Find the angle of elevation of the sun when a chimney 60 feet high throws a shadow 20√3 yards long. 7. Prove the identities: (1) (tan 0+2) (2 tan 6+1)=5 tan 0+2 sec2 0; 5п 8. One angle of a triangle is 45° and another is radians; 8 express the third angle both in sexagesimal and radian measure. 22 9. The number of degrees in an angle exceeds 14 times the number of radians in it by 51. Taking = find the sexagesimal measure of the angle. 10. 2 7 If B=30°, C=90°, b=6, find a, c, and the perpendicular from C on the hypotenuse. 12. The angle of elevation of the top of a pillar is 30°, and on approaching 20 feet nearer it is 60°: find the height of the pillar. 13. Shew that tan24 - sin2A=sin1A sec2A. 14. In a triangle the angle A is 3x degrees, the angle B is Пх a grades, and the angle C is radians: find the number of degrees in each of the angles. 300 15. Find the numerical value of sin3 60° cot 30° - 2 sec2 45° +3 cos 60° tan 45° – tan2 60°. 16. Prove the identities: (1) (1+tan 4)2+(1+cot A )2= (sec A+cosec A)2; (2) (seca - 1)2 — (tan a― sin a)2= (1 − cos a)2. 17. Which of the following statements is possible and which impossible? (1) cosec = a2+62 ; (2) 2 sin =a+ α 18. A balloon leaves the earth at the point A and rises at a uniform pace. At the end of 1.5 minutes an observer stationed at a distance of 660 feet from A finds the angular elevation of the balloon to be 60°; at what rate in miles per hour is the balloon rising? 19. Find the number of radians in the angles of a triangle which are in arithmetical progression, the least angle being 36°. 20. Shew that sin2 a sec2 B+tan2B cos2a=sin2a+tan2ß. 21. In the triangle ABC if A=42°, B=116° 33', find the perpendicular from Cupon AB produced; given |