By Art. 5, arc XnA = OX × 0, where is expressed in radians. Hence, using r for the radius and S for the area of sector, S = r20. Evidently the area of the segment XAn = S - area of triangle XOA. But area of triangle XOA= }0X.BA = } 0X . 04 sin 0 = r2 sin 0. Hence, using G for area of segment, As an exercise the student may later show that this formula holds when is an obtuse angle. Also when π < 0 < 2π. (See Art. 51.) EXERCISES (More difficult exercises for advanced work and review.) 1. A building 80 ft. long by 60 ft. wide has a roof inclined at 36° 45′ to the horizontal. Find the area of the roof, and show that the result is the same whether the roof does or does not have a ridge. Ans. 5990.6 sq. ft. 2. Find the width of the shadow of the wall shown in Fig. 49. If the height of the wall is h ft., the angle of elevation of the sun a, and the angle between the vertical plane through the sun and the plane of the wall, show that width of shadow = h cot a sin 0. 3. A wall extending east and west is 8 ft. high. clination of 49° 30′ and is 47° 15' 30" west of south. shadow of the wall. The sun has an inFind the width of the Ans. 4.637 ft. 4. A cliff 2000 ft. high is on the seashore; how far away is the horizon? What is the dip of the horizon? Ans. 54.77 mi.; 0° 47′ 33′′. 5. Find the greatest distance at which the lamp of a lighthouse can be seen from the deck of a ship. The lamp is 85 ft. above the surface of the water and the deck of the ship 30 ft. Ans. 18 mi. approx. 6. Find the radius of one's horizon if located 1250 ft. above the earth. How large when located 3 miles above the earth? Ans. 43.3 mi.; 154.17 mi. 7. How high above the earth must one be to see a point on the surface 50 miles away? Ans. 1666.7 ft. 8. If R and r are the radii of two pulleys, D the distance between the centers, and L the length of the belt, show that when the belt is not crossed, Fig. 50, the length is given by the following formula where the angle is taken in radians: 9. Using the same notation as in Exercise 8, show that when the belt is crossed, Fig. 51, the length is given by the following formula: L = 2 √D2 sin-1 R+r). D (R+r)2 + (R +r) ( x + 2 sin¬1 Note. These formulas would seldom be used in practice. An approximate formula would be more convenient, or the length would be measured with a tape line. A rule often given for finding the length of an uncrossed belt is: Add twice the distance between the centers of the shafts to half the sum of the circumferences of the two pulleys. D 10. Using the formula of Exercise 8, 12 ft., find the length of the belt. = rule. = and given R 18 in., r = 8 in., and Find the length by the approximate Ans. 30.87 ft.; 30.81 ft. 11. Use the same values as in Exercise 10, and find by the formula of Exercise 9 the length of the belt when crossed. Ans. 31.20 ft. 12. An open belt connects two pulleys of diameters 6 ft. and 2 ft. respectively. If the distance between their centers is 15 ft., find the length of the belt. Ans. 42.83 ft. 13. Two pulleys of diameters 7 ft. and 2 ft. respectively are connected by a crossed belt. If the centers of the pulleys are 16 ft. apart, find the length of the belt. Ans. 47.41 ft. 14. A ray of light after reflection at a plane mirror makes with the perpendicular to the mirror at the point of incidence an angle equal to the angle it makes with this perpendicular at incidence. Prove that if the mirror is turned through an angle a, the reflected ray is turned through an angle 2 a. 16. Compute the volume for each foot in the depth of a horizontal cylindrical oil tank of length 30 ft. and diameter 4 ft. 16. A cylindrical tank in a horizontal position is filled with water to within 10 in. of the top. Find the volume of the water if the tank is 10 ft. long and 4 ft. in diameter. Ans. 106.7 cu. ft. 17. Find the angle between the diagonal of a cube and one of the diagonals of a face which meets it. Ans. 35° 15.8'. 18. The slope of the roof in Fig. 52 is 33° 40'. Find the angle which is the inclination to the horizontal of the line AB, drawn in the roof and making an angle of 35° 20′ with the line of greatest slope. Ans. 26° 53' 14". 19. A hill slopes at an angle of 32° with the horizontal. A path leads up it making an angle of 47° 30′ with the line of steepest slope, find the inclination of the path with the horizontal. Ans. 20° 58′ 40′′. 20. Two roofs have their ridges at right angles, and each is inclined to the horizontal at an angle of 30°. Find the inclination of their line of intersection to the horizontal. 22. Show that placing the carpenter's square as shown in Fig. 54 (b) will determine " 4 P 5" FIG. 53. B the miter for making a regular pentagonal frame as shown in (a). What is the angle of the miter? 23. If 12 in. is taken on the tongue of the square, how many inches must be taken on the blade to cut miters for making regular polygons of the follow ing numbers of sides: 3, 4, 6, 8, and 10? Express results to the nearest 16th of an inch. Ans. 201; 12; 61†; 5; 37. 24. In the frame of a tower shown in Fig. 55, determine the distances from A and B, C and D, etc., to make the holes in the braces so that they may be bolted at points a, b, c, etc. These distances should be accurate to tenths of an inch. Can these distances be determined by means of geometry? Ans. Aa = 10 ft. 5.3 in., etc.; Yes. 25. What diameter of stock must be chosen so that a hexagonal end 21 in. across the flats may be milled upon it? Answer the same question for an octagon. The meaning of "across the flats" is shown in Fig. 56. Ans. 2.74 in.; 2.57 in. 26. A tripod is made of three sticks each 4 ft. long, by tying together the ends of the sticks, the other ends resting on the ground 21 ft. apart. Find the height of the tripod. Ans. 3 ft. 8 in. sin @ CHAPTER IV GRAPHICAL REPRESENTATION OF TRIGONOMETRIC FUNCTIONS 43. Line representation of the trigonometric functions.— Construct a circle of radius OH, with its center at the origin of coördinates, Fig. 57. Since, in finding the trigonometric functions of an angle with its vertex at the origin of coördinates and its H tan-0 M (b) -tan-0 cot. Y E F D initial side on the positive part of the axis of abscissas, any point may be chosen in the terminal side of the angle, we may take the point where the terminal side cuts the circumference of the circle. Draw angle angle XOP in each of the four quadrants, and draw MP10X in each case. Now choose OH as the unit of measure, that is, OH 1. Then in each of the four quadrants, |