PROPOSITION IV. To express the cosine of an angle of a plane triangle in terms of the sides of the triangle. Let ABC be a triangle ; A, B, C, the three angles; a, b, c, the corresponding sides. 1. Let the proposed (A) be acute. From C draw CD perpendicular to AB, the base of the triangle. Then, BC=AC'+AB-2AB. AD (Prop. XXVI. B. IV. El. Geom.) Or, a2 = b2 + c2 —2c. AD But, since CDA is a right-angled triangle, = C 2 Let the proposed angle (A) be abtuse. From C, draw CD perpendicular to AB producea. Then, Or, BC AC + AB2 + 2AB.AD a2 = b2 + c2 + 2c AD But, since CDA is a right angled triangle, AD AC cos. CAD DA It will be seen that this result is identical with that which we deduced in the last case, so that, whether A be acute or obtuse, we shall have, To express the sine of an angle of a plane triangle in terms of the sides of the triangle. Let A be the proposed angle; then by last prop., cos. A = Adding unity to b2 + c2 a2 2bc - each member of the equation, 1 + cos. A = 1+ 2bc Multiplying together equations (1) and (2,) (1+cosA). (1-cosA)= But (1+cos. A) (1-cos. A)= 1-cos.' A = sin.' A (Table I.) .. sin. A = (a+b+c) (b+c-a) (a+c-b) (a+b—c) 462 c2 (a+b+c) (b+c—a) (a+c—b) (a+b—c) Extracting the root on both sides, sin. A = 1 4b2c2 2bc √(a+b+c)(b+c—a)(a+c—b)(a+b—c)..(3 The above expresison, for the sine of an angle of a triangle in terms of the sides, is sometimes exhibited under a form some what different. Let s denote the semiperimeter, that is to say, half the sum of the sides of the triangle; then Substituting 2 s, 2 (sa),...... for a+b+c, b+c-a,..... in the expression for sin.' A, it becomes sin.' A = 16 s (s—a) (s—b) (s—c) 4 b'c2 And extracting the root on both sides, 2 sin. A = (s—c) bc⋅ √ ́s (s—a) (s—b) (s Proceeding in the same manner for the other angles, we shall find (8) Dividing the formulæ marked (3) by those marked (0) BEFORE proceeding to apply the formulæ deduced in the last chapter to the solution of triangles, we shall make a few remarks upon the construction of those tables, by means of which we are enabled to reduce our trigonometrical calculations to numerical results. It is manifest, from definitions 1°, 2°, 3°, &c. that the various trigonometrical quantities, the sine, the cosine, the tangent, &c. are abstract numbers representing the comparative length of certain lines. We have already obtained the numerical value of these quantities in a few particular cases, and we shall now show how the numbers, corresponding to angles of every degree of magnitude, may be obtained by the application of the most simple principles. The numbers corresponding to the sine, cosine, &c. of all angles from 1" up to 90°, when arranged in a table, form what is called the Trigonometrical canon. The first operation to be performed is To compute the numerical value of the sine and cosine of 1'. We have seen, Chap. II. formula (j) that By which formula the sine of any angle is given in terms of the sine of twice that angle. |