235. If a, ẞ, y, & be orts, then from the identities (iii) and (iv) of the last paragraph we obtain α Fig. 148. sin By. sin að + sin ya. sin 38+ sin aß. sin yồ = 0...(1), sin By. cos ad + sin ya. cos Bồ + sin aß. cos yồ: = 0...(2); trigonometrical identities between the sines and cosines of the angles between any four lines in a plane. (In order that the angles between the lines may be perfectly definite it is necessary to give the lines senses.) 236. From the last two identities obtained, a great number of well known trigonometrical formulae are at once derived by giving special values to one of the orts (say 8). We will give a few examples. (a) Let &= ẞ, aß = A, By = B, then or Substituting in (ii) of § 235 we obtain (b) then and sin B. cos A - sin (A + B) + sin A. cos B = 0, sin A ·B= sin A. cos B – sin B. cos A. (c) Let (g) Let By=A, âd=B, ßd=71⁄2, then from (i) sin A. sin B+sin(A+B -sin (B) sin (-4)=0, 2 cos (A + B) = cos A. cos B-sin A . sin B. and hence then cos (AB) = cos A. cos B + sin A . sin B. 237. The relations between the sides and angles of a triangle are also easily obtained by means of vectors. Let ABC be any triangle, A, B, C the measures of the angles, a, b, c the sides opposite them, a, B, y the sides as vectors. Again, multiply (1) by a taking the scalar product, we get or which is the same as or a2 = ac cos B + ab cos C, a = c cos B+ a cos C.... .(3). Multiply (1) by a, taking the vector product, we get For a right-angled triangle having C as the right angle, (a) = 0 and the magnitude of [a] is ab, hence (2) becomes c2 = a + b2, (3) becomes a = c cos B, (4) becomes c sin B = b. 238. It would be beyond the scope of the present work to enter into the details of trigonometrical problems. Sufficient however has been done to shew that VectorAlgebra does implicitly contain Trigonometry. EXERCISES XIV. (1) If a and B are orts, prove that (aß)2+[aß]2 = 1, and hence that cos2 A+ sin2 A = 1. (2) Prove directly from vectors the relation between the sides and between the sides and angles of a right-angled triangle. (3) If a, ß, y are orts, aẞ= A, By=B, ây π = 2' (4) If a, B, γ be three orts in a plane, prove directly from the relation y= Aa+ BB the expansion for sin (A – B), cos (A+B), and cos (A-B). (5) Prove directly by vectors that sin 242 sin A cos A, cos 2A=cos2 A-sin2 A. (6) From the vector theorem shew that ([By][ad])+([ya][B8])+([aß][yd])=0, sin (A+B) sin (A - B)=sin2 A - sin2 B (7) Find directly from vectors the radii of the circumscribing and inscribed circles of a triangle. (8) Shew that the area of any quadrilateral is measured by the product of the diagonals and the sine of the included angle. CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. |