the positive x-axis, and the hypotenuse AB lies in the first quadrant. Then the definitions on page 166 show that sin A= a/c, cos A b/c, tan A = a/b\ = = c/a (1) It is not always convenient to place the triangle on coördinate axes, and sometimes other letters must be used, and hence it is desirable to remember these formulas in words, as follows: The sine of an acute angle of a right triangle is the ratio of the side opposite the angle to the hypotenuse. The cosine of an acute angle of a right triangle is the ratio of the side adjacent to the angle to the hypotenuse. The tangent of an acute angle of a right triangle is the ratio of the side opposite the angle to the adjacent side. Analogous statements for the last three functions are readily obtained by the reciprocal relations (page 166). The sides and angles of a triangle are called its parts. In order to construct a right triangle, we must be given two parts, in addition to the right angle, of which at least one must be a side. To solve a triangle is to find the unknown parts from the known. Every right triangle may be solved by means of formulas (1) and the fact that A + B = 90°. But the Pythagorean Theorem is sometimes convenient. In solving right triangles but two essentially different cases arise: I. Given a side and an angle, to find the other two sides use those two of equations (1) which contain the unknown sides in the numerators and the given side in the denominator. II. Given two sides, to find an angle, use that one of the two equations (1) containing the given sides which leads to the simpler division, and then to find the third side use either of the two equations containing the third side in the numerator. In either case the second angle is found from A + B 90°. The use of the equations indicated in these rules will lead to multiplication, and division by a value of sin 0, for example, is avoided. = It is desirable that figures be constructed accurately, as the figure may show the absurdity of an incorrect result. A good check on the accuracy of the computation may be obtained by constructing the triangle with scale and protractor, and measuring the unknown parts. EXAMPLE 1. Solve the right triangle A = 37°.24, b = 9. = sec A, whence c = FIG. 100. Check. The accuracy of the computation may be checked by finding b from a and c. By the Pythagorean Theorem, and tables of squares and square roots, = 9. which agrees with the given value of b. EXAMPLE 2. Solve the right triangle a = 7, b The given sides occur in the third of formulas (1). Hence 0.7778, whence A = 37°.87, and hence B = 90° – 37°.87 To find c we have whence c = a csc A B It is to be noted that the angles and lines in these examples and in the exercises following have not been measured with the same degree of precision. The angles are given to four figures in order to afford practice in interpolation. Problems involving isosceles triangles and regular polygons may be solved by means of right triangles, for such figures may be divided into congruent right triangles. 65. Applications. An instrument known as a transit enables surveyors to measure angles in vertical and horizontal planes. C If A and B are points not in the same horizontal plane, and if AC and BD are horizontal lines in the vertical plane through A and B, then BAC is called the angle of depression of B at A, and ZABD is called the angle of elevation of A at B. If the eyes are at A, looking horizontally over B, the angle of depression of B is the angle through which the eyes must be lowered to see B. While if the eyes are at B, looking horizontally below A, the angle of elevation of A is the angle through which the eyes must be raised to see A. D B FIG. 102. If AA' and BB' are the vertical lines through two points A and B, meeting the horizontal plane through a third point C A B B' at A' and B' respectively, then ZA'CB' is called the horizontal angle between A and B at C. If we think of AA' and BB' as two trees, of different heights usually, for an observer at C, the horizontal angle between the tree tops A and B is the angle through which one must turn, if one faces first toward the tree AA' and then turns to face the tree BB'. C FIG. 103. The bearing of a line is the angle the line makes with some fundamental line of the figure which is called the base line. For example, if the base line is north and south, the bearing of a line running northeast is 45° east of north. EXERCISES 1. In any right triangle, if c and A are given, show that a = c sin A and b = c cos A. Express these equations in words. If a and A are given, find b and c. If b and A are given, find a and c. 3. To find the width of a river, two points, A and C, are taken on one bank 100 feet apart. If B is the point on the other bank directly opposite C, and if ZCAB equals 72°.16, how wide is the river? 4. From the top of a lighthouse 35 feet high, the angle of depression of a ship is 11°.38. How far is the ship from the lighthouse? 5. What is the angle of elevation of the sun if a pole 49 feet high casts a shadow of 11 feet? 6. An iron wedge for splitting rails is to have a base two inches wide and a vertex angle of 15°. How long will each side be? 7. A rustic summer house, or shelter, is to be built with an octagonal floor, 6 feet on a side. Determine the amount of flooring necessary, making an allowance of 25% on account of the “tongue and groove" and for waste. 8. Solve Exercise 7 if the floor is to be pentagonal. Solve Exercise 7 if the floor is to be a regular seven-sided polygon. Can this Exercise be solved by plane geometry? 9. If the radius of a regular polygon of n sides is r, find (a) the perimeter in terms of n and r; (b) the area. 10. If a side of a regular polygon is a, express the area as a function of a, assuming n to be constant. 11. What is the angle of inclination of an ordinary gable roof, if its pitch (the ratio of its height to its entire width) is ?? ? 12. A point P moves with uniform speed around a circle of radius one foot, making a complete revolution every 36 seconds. The projection M of P on a diameter AOB, moves along the diameter. Find OM for the values 50°, 60°, 70°, if 0 = LAOP. Find the average velocity P FIG. 104. 13. Two life saving stations are 10 miles apart on a beach running 22°.5, east of north. north of east from one How long would it take A lightship anchored off the beach lies 33°.75 station, and 11°.25 east of south from the other. a boat to run from one station to the other if its speed is 10 miles per hour, and if it passes outside the lightship? How long would it take the boat to go from the ship to the nearest point on the beach? 14. (a) Two sides of a triangle, not a right triangle, are 12 and 20, and the included angle is 28°.48. Find the altitude on the latter side and the area. (b) Find the area of any triangle ABC in terms of the sides b and c and the included angle A. 15. Find the area of a parallelogram in terms of two adjacent sides and the included angle. 16. Two ships are on a line with a lighthouse, which is 40 feet high. At the top of the lighthouse, the angles of depression of the ships are respectively 4°.26 and 6°.31. How far apart are the ships? 17. A regular octagonal tower is 4 feet on each side. The roof is 10 feet high. How long should the rafters be? 18. A bridge is to be built across a ravine between two points A and B on the same level. Two stations C and D are chosen in the ravine which lie in the vertical plane through A and B. At A, the angle of depression of C is 42°.37, and AC 50 feet. At C the angle of depression of D is 4°.52, and CD 40 feet. At D the angle of elevation of B is 37°.89, and Find the length of the bridge. DB = 60 feet. = 19. Check the accuracy of the measurements in Exercise 18, by finding the difference in altitude of A and D, and also of B and D. 20. To find the width of a river, a tree is selected on one shore. At a point 50 feet from the tree, the angle of elevation of the top is 48°.72. At the point on the other shore directly opposite the tree, the angle of elevation of the top is 17°.39. How wide is the river? 64°.37870.24 1000 ft. B FIG. 105. 21. From a station A the horizontal angle between a mountain top C and a second station B, 1000 feet from A and at the same altitude, is 64°.37. At B the horizontal angle between the mountain top and A is 90°, and the angle of elevation of the mountain top is 37°.24. How high is the mountain? 66. Parallelogram Law - Velocities, Accelerations, Forces. The law to be considered in this section is illustrated in B EXAMPLE 1. The current in a river flows at the rate of 2 miles an hour. A man rows across at the rate of 4 miles an hour, keeping his boat at right angles to the shore. Show that the boat moves in a straight line. Find how fast it moves, and in what direction. Take the starting point for the origin of a system of coördinates, and let the x-axis lie along the bank of a river. At the time t, the boat will be at a point P, whose coördinates give the distance the boat has been carried by the current, x = 2t, and the distance the man has rowed from shore, y = 4t. Eliminating t, we A.M |