If the other angles are required, they may be found by Case 1st. 2. In a triangle ABC, are given AB and BC = 34, to find the angle B. Ans. 38. 3. In a triangle ABC are given AC = 88, AB = 108, and BC = 54, to find the angle C. Ans. C=96° 4′. The preceding rules solve all the cases of plane triangles, both right-angled and oblique. There are however other rules, suited to right angled triangles, which are sometimes more convenient than the general ones. Previous to giving these rules, it will be necessary to make the following Remarks on right-angled triangles. 1. ABC, Fig. 50, being a right-angled triangle, make one leg AB radius, that is, with the centre A, and distance AB, describe an arc, BF. Then it is evident that the other leg BC represents the tangent of the arc BF, or of the angle A, and the hypothenuse AC the secant of it. 2. In like manner, if the leg BC, Fig. 51, be made radius; then the other leg AB will represent the tangent, of the arc BG, or angle C, and the hypothenuse AC the secant of it. (AC + AB + BC) × ( (AC + AB + BC) - BC); therefore AB X AC: (AC + AB + BC) × ( (AC + AB+ BC) - BC) :: rad.2: (cos. BAC)2. Now it is evident, that in working this proportion by logarithms, and taking the arithmetical complements of the logarithms of the first term, viz. of the two sides including the required angle, if we omit the logarithm of the square of radius, which is 20, it is just equivalent to rejecting 20 from the sum of the logarithms; which 3. But if the hypothenuse be made radius; then each leg will represent the sine of its opposite angle; namely, the leg AB, Fig. 52, the sine of the arc AE or angle C, and the leg BC the sine of the arc CD, or angle A. The angles and one side of a right-angled triangle being given to find the other sides. RULE. Call any one of the sides radius, and write upon it the word radius; observe whether the other sides become sines, tangents, or secants, and write these words on them accordingly. Call the word written upon each side the name of that side. Then, As the name of the side given, Is to the name of the side required; To the side required.* Two sides of a right-angled triangle being given, to find the angles and other side. RULE. Call either of the given sides radius, and write on them as before. Then, DEMONSTRATION. Let ABC, Fig. 53, be a right-angled triangle; then it is evident that BC is the tangent, and AC the secant of the angle A, to the radius AB. Let AD represent the radius of the tables, and draw DE perpendicular to AD meeting AC produced in E; then DE is the tangent, and AE the secant of the angle A, to the radius AD. But because of the similar triangles ADE, ABC, AD: DE:: AB: BC; that is the tabular radius: tabular tangent :: AB : BC. Also AD: AE::AB: AC; that is, the tabular radius: tabular secant :: AB: AC. These proportions correspond with the rule. When either As the side made radius, Is to the other given side; So is radius, To the name of that other side.* After finding the angle, the other side is found as in the preceding rule. EXAMPLES. I. In a right-angled triangle ABC, are given the base AB = 208, and the angle A = 35° 16′, to find the hypothenuse AC and perpendicular BC. |