Then, in the triangle APC, right angled at P, Whence the ZA=37 20', the B-27° 04', and the C 115° 36'. INSTRUMENTALLY. In the first proportion, extend from 345 to 406, on the line of numbers; that extent will reach, upon the same line, from 58 to 68'2, the difference of the segments of the base. In the second proportion, extend from 174 to 138 on the numbers; that will reach, on the sines, from 90° to 52°. In the third proportion, extend from 232 to 206, and that extent will reach from 90° to 63°. NOTE 1. These three problems include all the cases of plane triangles, as well right-angled as oblique-angled. There are some other theorems, suited to some particular forms of triangles, which are often more expeditious in practice than the preceding general methods. One of which, as the case, for which it serves, so often occurs, is here given. PROBLEM. Given the angles and a leg of a right-angled triangle; to find the other leg and the hypotenuse. Given EXAMPLE. In the plane triangle ABC, right-angled at B, AB 162 {2 350 07′ 48′′} Required AC and BC. GEOMETRICALLY. Make AB=162, and the angle A=53° 07′ 48′′; then raise the perpendicular BC meeting AC in C. So shall AC measure 270, and BC 216. * DEMONSTRATION. With the centre A and any radius AD, describe an arc DE, and erect the perpendicular DF; which, it is evident, will be the tangent, and AF the secant of the arc DE, or angle A, to the ra. dius AD. And in similar triangles ADF, ABC, it will be AD: AB:: DF: BC:: AF: AC. Q. E. D. The extent from 45° to 53° 08', upon the tangents, will reach from 162 to 216 upon the numbers. NOTE 2. It is common to add another method for right-angled triangles, which is this. ABC being the triangle, make a leg AB radius, that is, with centre A and radius AB, describe an arc BF: then it is evident, that the other leg BC represents the tangent, and the hypotenuse AC the secant of of the angle A or arc BF. In like manner, if the leg BC be made radius, then the leg AB will represent the tangent, and AC the secant of the arc BG, or the angle C. But if the hypotenuse be made radius, then each leg will represent the sine of its opposite angle; namely, the leg AB Ccc |