1. EXAMPLES. VII. c. 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 1.625 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 1.9 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? π= 22 7. The large hand of a clock is 2 ft. 4 in. long; how many inches does its extremity move in 20 minutes? π= 22 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 = and the earth's diameter as 7920 miles. " 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 1° may be 1 foot, taking π= 11 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 19. 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 0+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 sexa, 7 gesimal measure of the angle. 10. 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 tan2A - sin2A=sin1A sec2A. 14. In a triangle the angle A is 3x degrees, the angle B is x 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 A)2+(1+cot A)2=(sec A+cosec A)2; (2) (sec a -1)2— (tan a― sin a)2=(1 − cos a)2. 17. Which of the following statements is possible and which impossible? 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 sec2ß+tan2B cos2a=sin2a+tan2ß. 21. In the triangle ABC if A=42°, B=116°33′, find the perpendicular from Cupon AB produced; given c=55, tan 42°='9, tan 63° 27' = 2. (2) cosec a (sec a-1)- cot a (1-cos a)=tan a-sin a. 23. Shew that 1+cot 60°\2 1+ cos 30° = 1- cos 30°° 24. A man walking N.W. sees a windmill which bears N. 15° W. In half-an-hour he reaches a place which he knows to be W. 15°S. of the windmill and a mile away from it. Find his rate of walking and his distance from the windmill at the first observation. 25. Find the number of radians in the complement of 26. Solve the equations : (1) 3 sin 0+4 cos20=4}; (2) tan 0+ sec 30°=cot 0. 27. If 5 tan a=4, find the value of Зп 8 29. Find the distance of an observer from the top of a cliff which is 1952 yards high, given that the angle of elevation is 77° 26', and that sin 77° 26'='976. 30. A horse is tethered to a stake by a rope 27 feet long. If the horse moves along the circumference of a circle always keeping the rope tight, find how far it will have gone when the rope has traced out an angle of 70°. π= 22 CHAPTER VIII. TRIGONOMETRICAL RATIOS OF ANGLES OF ANY 72. In the present chapter we shall find it necessary to take account not only of the magnitude of straight lines, but also of the direction in which they are measured. Let O be a fixed point in a horizontal line XX', then the position of any other point P in the line, whose distance from 0 is a given length a, will not be determined unless we know on which side of O the point P lies. But there will be no ambiguity if it is agreed that distances measured in one direction are positive and distances measured in the opposite direction are negative. Hence the following Convention of Signs is adopted: lines measured from 0 to the right are positive, Thus in the above figure, if P and Q are two points on the line XX' at a distance a from O, their positions are indicated by the statements i OP=+a, OQ= — a. 73. A similar convention of signs is used in the case of a plane surface. Let O be any point in the plane; through O draw two straight lines XX' and Y' in the horizontal and vertical direction respectively, thus dividing the plane into four quadrants. |