Solving the first two equations for 0 and 4, by adding and subtracting, and dividing by 2, we have 0 = 1 (a + B), & = 1⁄2 (a − ß), sin a + sin ẞ = 2 sin (a + ẞ) cos (a – ẞ). and hence, substituting, (1) (a B). (3) B = ß). (4) sin a sin ẞ 2 cos(a + B) sin cos a + cos B = 2 cos(a + B) cos cos acos ẞ - 2 sin (a + ẞ) sin (a – ẞ). Equations (1) and (2) give properties of the function sin x. What are the analogous properties of log x? To what property of 22 is (2) analogous? 125. Logarithmic Solution of Triangles, Case III. The law of sines (page 201) may be written From (6), (1), and (4), page 333, we then have If two sides and the included angle of a triangle are given, the other angles may be found by means of (1), as is the Check. Find one of the given sides from c, A or B, and C. 126. Logarithmic Solution of Triangles, Case IV. We seek an expression for tan A, A being an angle of a triangle ABC, in terms of the sides. We have Interchanging the values of a, b, c, does not affect the values It is proved in most elementary geometries that the area of a triangle in terms of its sides is = 34.25, b = 42.91, A simpler proof is indicated in Exercises 6 and 7 below. The angles are found by (1) and (2), the first step in the computation being to find log r from its value in (1). 3. Solve and check the following triangles, and find the areas. 4. Two men start to walk from the same point in directions inclined at 45° to each other. When one has gone 792 yards, the other has gone 846 yards. What is the inclination of the line joining them to their paths, and how far apart are they? 5. The sides of a triangular field are 84.3 rods, 77.5 rods, and 102.1 rods. What are the angles of the field? How many potatoes could be raised on it, if the yield were 200 bushels per acre? 8. Find the diameter of the circle circumscribed about a triangle. Hint: Draw the diameter CD through C, and join D to B. What are the 21.5 rods. The angle between the first two sides is 84°.5. How many acres does the field contain? |