Figure 1-26.-Any triangle, three sides given.Figure 1-27.-0blique triangle (law of tangents).For any pair of sides—as side a and sideb—the law may be expressed as follows:For the triangle shown in figure 1-27, youknow the lengths of two sides and the size of theangle between them. You can determine the sizesof the other two angles by applying the law oftangents as follows.First note that you can determine the value ofangles (B + C), because (B + C) obviouslyequals 180° – A, or 180° – 34°, or 146°. Now,if you know the sum of two values and thedifference between the same two, you candetermine each of the values as follows:Now, you know the sum of (B + C).Therefore, if you could determine the difference,or (B – C), you could determine the sizes of Band C You can determine 12(B — C) from thelaw of tangents, written as follows:One-half of (B + C) means one-half of 146°,or 73°. The tangent of 730 is 3.27085. Thesolution for 12(B – C) is therefore as follows:(from table of natural tangents) 1/2 (B - C)= 19°58’ (B – C) = 2(19058’) = 39°56’Knowing both the sum (B + C) and thedifference (B – C), you can now determine thesizes of B and C as follows:The Ambiguous CaseWhen the given data for a triangle consists oftwo sides and the angle opposite one of them, itmay be the case that there are two triangles thatconform to the data. A situation in which therecan be two triangles is called the ambiguous case.Figure 1-28 shows two possible triangles thatFigure 1-28.-Two ambiguous case triangles (solution of onewill satisfy the other).1-23
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