(9) Given L cosec 46.23' = 10·1402787, L cosec 46.24' = 10.1401584, find the angle whose L cosec is 10.1402567. CHAPTER XIV. ON THE RELATIONS BETWEEN THE SIDES OF A TRIANGLE AND THE TRIGONOMETRICAL RATIOS OF THE ANGLES OF THE TRIANGLE. 172. A TRIANGLE is composed of six parts, three sides and three angles. Three of these parts being given, one at least of the three being a side, we can generally determine the other three parts. If only the three angles be given, we cannot determine the sides, because an infinite number of triangles may be constructed with the three angles of the one equal to the three angles of the other, each to each. 173. We shall denote the angles of a triangle by the letters A, B, C; the sides respectively opposite to them by the letters a, b, c. The student must remember the results established in Art. 97, sin (180°-4)=sin A, Thus, if ACB be an exterior angle of the triangle ACD, The results of Art. 62 are also frequently employed in this and the next Chapters. Let A, B be any two angles of the triangle ABC, and as one of them must be acute, let it be A. Then according as B is acute or obtuse, draw CD at right angles to AB or to AB produced. If the angle at B be a right angle the theorem holds good, for then cos B = cos 90o = 0, and .'. a. cos B = 0, and we have c = b.cos A. 175. To shew that in every triangle the sides are proportional to the sines of the opposite angles. Let A, B be any two angles of the triangle ABC, and as one of them must be an acute angle, let it be A. Then, according as B is acute or obtuse, draw CD at right angles to AB or to AB produced. If the angle at B be a right angle the theorem still holds good, 176. To express the cosine of an angle of a triangle in terms of the sides. Let A, B be any two angles of the triangle ABC, and as one of them must be acute, let it be A. Then, according as B is acute or obtuse, draw CD at right angles to AB or to AB produced. or Now, in fig. 1, by Euclid II. 13, ACAB+ BC-2AB. BD, b3 = c2 + a2 - 2c. BD. Now, from the right-angled triangle BCD, Again, in fig. 2, by Euclid II. 12, AC® = AB* + BC +2AB.BD, b3 = c2 + a2 + 2c. BD. |