Trigonometry, with the Theory and Use of Logarithms

Trigonometry, with the Theory and Use of Logarithms

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 edition. Excerpt: ...solution just given, is by means of the Law of Tangents. 6-c _ tan I (B-C) b + c tan (B + C)' We explain this method by means of the same numerical example as was used to illustrate the last method. We first compute b--c, b + c, iCft+cy. Check by the formula a sin B = b sin A. 39. Case IV. Three Sides. As in Case III, triangles under Case IV may always be solved by the Law of Cosines. From c2 = a2 + b2--2 ab cos C we get a2 + 62 _ CsC=--2ab--' which is a form from which the angle may be computed. Given: a = 51.00 6 = 65.00 c= 20.00 aa = 2601 2ab = 6630 62 = 4225 2 5c = 2600 c2 = 400.0 2a? = 2040 C=18Q-(A + B) = Check by means of the Law of Sines. Or we may compute C directly by the method just used for A and B, and use the relation A + B + C= 180 as a check. If the numbers cannot be conveniently squared, the triangle may be solved by formulas better adapted to logarithmic computation. These formulas may be derived as follows: The Half-angle Formulas. By dividing by one another formulas (6) and (7) of 27, we find where the positive sign is used before the radical because, A being an angle of a triangle and therefore less than 180, A is necessarily acute, and therefore has a positive tangent. We wish now to express the radical in terms of the sides of the triangle. For this purpose we use the Law of Cosines: cos A------2 bc Hence 1-cos A =-c = (a b c(g = b + c, 2 bc 2 be' These values substituted in (1) give the result we wish. The result may, however, be put in a more convenient form by means of the notation (2) a + b + c = 2 8. We have then a + b--c = 2(--c), a--b + c=2(s--6), b + c--a = 2(s--a). Hence 1-cos A 2(s-b8-c, 1 + cos A =--'-be These values substituted in (1) give the formula s(s--a)' from which, by a more

Product details

  • Paperback | 34 pages
  • 189 x 246 x 2mm | 82g
  • United States
  • English
  • black & white illustrations
  • 1236950186
  • 9781236950185