24. Two men, M and N, 3200 ft. apart, observe an aeroplane A at the same instant, and at a time when the plane MNA is vertical. The angle of elevation at M is 41° 27' and the angle at N is 61° 42'. Required AB, the height of the aeroplane. Show that h cot 41° 27′+ h cot 61° 42′ is known, whence h can be found. M 25. A kite string 475 ft. long makes an angle of elevation of 49° 40'. Assuming the string to be straight, what is the altitude of the kite? = F E 26. A steel bridge has a truss ADEF in which it is given that AD = 20 ft., BF = 6 ft. 8 in., and FE : 12 ft., as shown in the figure. Required the angle of slope which AF makes with AD. 27. Two tangents are drawn from a point P to a circle and contain an angle of 37.4°. The radius of the circle is 5 in. Find the length of each tangent and the distance of P from the center. 28. From the top of a cliff 95 ft. high, the angles of depression of two boats at sea are observed, by the aid of an instrument 5 ft. above the ground, to be 45° and 30° respectively. The boats are in a straight line with a point at the foot of the cliff directly beneath the observer. What is the distance between the boats? 29. A carpenter's square BCA is held against the vertical stick BD resting on a sloping roof AD, as in the figure. It is found that AC 24 in. and CD=11.5 in. Find the = angle of slope of the roof with the horizontal. 30. In Ex. 29 find the length of AD. 31. A man 6 ft. tall stands 4 ft. 9 in. from a street lamp. If the length of his shadow is 19 ft., how high is the light above the street? 32. The shadow of a city building is observed to be 100 ft. long, and at the same time the shadow B of a lamp-post 9 ft. high is observed to be 5.2 ft. long. Find the angle of elevation of the sun and the height of the building. 33. A man 5 ft. 10 in. tall walks along a straight line that passes at a distance of 2 ft. 9 in. from a street light. If the light is 9 ft. 6 in. above the ground, find the length of the man's shadow when his distance from the point on his path that is nearest to the lamp is 3 ft. 8 in. 34. A man on a bridge 35 ft. above a stream, using an instrument 5 ft. high, sees a rowboat at an angle of depression of 27° 30'. If the boat is approaching at the rate of 23 mi. an hour, in how many seconds will it reach the bridge? 35. A shaft 0, of diameter 4 in., makes 480 revolutions per minute. If the point P starts on the horizontal line 04. how far is it above OA after of a second? 1 48 36. Assuming the earth to be a sphere with radius 3957 mi., find the radius of the circle of latitude which passes through a place in latitude 47° 27′ 10′′ N. 37. When a hoisting crane AB, 28 ft. long, makes an angle of 23° with the horizontal AC, what is the length of AC? Suppose that the angle CAB is doubled, what is then the length of AC? 38. In Ex. 37 find the length of BC in each of the two cases. 39. Wishing to measure the distance AB, a man swings a 100-foot tape line about B, describing an are on the ground, and then does the same about A. The arcs intersect at C, and the angle ACB is found to be 32° 10'. What is the length of AB? B 40. From the top of a mountain 15,250 ft. high, overlooking the sea to the south, over how many minutes of latitude can a person see if he looks southward? Use the assumption stated in Ex. 36. 41. The length of each blade of a pair of shears, from the screw to the point, is 51 in. When the points of the open shears are 37 in. apart, what angle do the blades make with each other? 42. In Ex. 41 how far apart are the points when the blades make an angle of 28° 45' with each other? 43. The wheel here represented has eight spokes, each being 19 in. long. How far is it from A to B? from B to D? 44. The angle of elevation of a balloon from a station directly south of it is 60°. From a second station lying. 5280 ft. directly west of the first one the angle of elevation is 45°. The instrument being 5 ft. above the level of the ground, what is the height of the balloon? CHAPTER V TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 71. Need for Oblique Angles. We have thus far considered only right triangles, or triangles which can readily be cut into right triangles for purposes of solution. There are, however, oblique triangles which cannot conveniently be solved by merely separating them into right triangles. We are therefore led to consider the functions of oblique angles, and to enlarge our idea of angles so as to include angles greater than 180°, angles greater than 360°, and even negative angles and the angle 0°. 72. Positive and Negative Angles. We have learned in algebra that we may distinguish between two lines which extend in opposite directions by calling one positive and the other negative. For example, in the annexed figure we consider OX as positive and therefore OX' as negative. We also consider OY as positive and hence OY' as negative. In general, horizontal lines extending to the right of a point which we select as zero are considered positive, and those to the left negative. Vertical lines extending upward from zero are considered positive, and those extending downward are considered negative. + + -X With respect to angles, an angle is considered positive if the rotating line which describes it moves counterclockwise, that is, in the direction opposite to that taken by the hands of a clock. An angle is considered negative if the rotat B ing line moves clockwise, that is, in the same direction as that taken by the hands of a clock. A Β' Arcs which subtend positive angles are considered positive, and arcs which subtend negative angles are considered negative. Thus ZAOB and are AB are considered positive; AOB' and arc AB' are considered negative. For example, we may think of a pendulum as swinging through a positive angle when it swings to the right, and through a negative angle when it swings to the left. We may also think of an angle of elevation as positive and an angle of depression as negative, if it appears to be advantageous to do so in the solution of a problem. 73. Coördinates of a Point. In trigonometry, as in work with graphs in algebra, we locate a point in a plane by means of its distances from two perpendicular lines. These lines are lettered XX' and YY', and their point of intersection 0. The lines are called the axes and the point of intersection the origin. In some branches of mathematics it is more convenient to use oblique axes, but in trigonometry rectangular axes are used as here shown. to represent one unit, in which case P = (4, 3), because its abscissa is 4 and its ordinate 3. Following a helpful European custom, the points are indicated by small circles, so as to show more clearly when a line is drawn through them. The abscissa and ordinate of a point are together called the coördinates of the point. 74. Signs of the Coördinates. From § 73 we see that abscissas to the right of the y-axis are positive; abscissas to the left of the y-axis are negative; ordinates above the x-axis are positive; ordinates below the x-axis are negative. A point on the line YY' has zero for its abscissa, and hence the abscissa may be considered as either positive or negative and may be indicated by ±0. Similarly, a point on the line XX' has 0 for its ordinate. 75. The Four Quadrants. The axes divide the plane into four parts known as quadrants. Because angles are generally considered as generated by the rotating line moving counterclockwise, the four quadrants are named in a counterclockwise order. Quadrant XOY is spoken of as the first quadrant, YOX' as the second quadrant, X'OY' as the third quadrant, and Y'OX as the fourth quadrant. 76. Signs of the Coördinates in the Several Quadrants. From § 74 we have the following rule of signs: In quadrant I the abscissa is positive, the ordinate positive; 77. Plotting a Point. Locating a point, having given its coördinates, is called plotting the point. For example, in the first of these figures the point (2, 4) is shown in quadrant II, the point (-3, -2) in quadrant III, and the point (1, 1) in quadrant IV. In the second figure the point (-2, 0) is shown on OX', between quadrants II and III, and the point (1, 0) on OX, between quadrants I and IV. In the third figure the point (0, 1) is shown on OY, between quadrants I and II, and the point (0,3) on OY', between quadrants III and IV. In every case the origin O may be designated as the point (0,0). 78. Distance from the Origin. The coördinates of P being x and y, we may form a right triangle the hypotenuse of which is the distance of P from 0. Representing OP by r, we have r = √x2 + y2. Since this may be written r = ± √x2+ y2, we see that r may be considered as either positive or negative. It is the custom, however, to consider the rotating line which forms the angle as positive. If r is produced through O, the production is considered as negative. 1. What is the distance of the point (3, 4) from the origin? r = √32 + 42 = √25 = 5. 2. What is the distance of the point (-3, 2) from the origin? - r = √(− 3)2 + (− 2)2 =V9+ 4 = √13 = 3.61. 3. What is the distance of the point (5,-5) from the origin? 4. What is the distance of the point (-2, 0) from the origin? r = √(− 2)2 + 02 = √4 = 2, as is evident from the conditions of the problem. |