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Solving Oblique Triangles

Problems

The Ambiguous Case

The Law of Cosines

Problem : In Triangle ABC, a = 4 , b = 3 , and B = 122 o . Is a triangle determined? If so, how many?

No. No such triangle exists.

Problem : If the side opposite the given angle is longer than the other given side, how many triangles are determined?

One.

Problem : Solve Triangle ABC given that a = 12 , b = 7 , and B = 36 o .

sin(A) = 1.07 . No solution. Sine never exceeds one.

Problem : Solve Triangle ABC given that a = 7 , b = 6 , and B = 45 o .

sin(A) = .82 . A 55.6 o or 124.4 o . This is an example of case three discussed in the text. The first possible triangle, an acute triangle, has parts a = 7 , b = 6 , c 8.3 , A 55.6 o , B = 45 o , C 79.4 o . The second possible triangle, and obtuse triangle, has parts a = 7 , b = 6 , c 1.6 , A 124.4 o , B = 45 o , and C 10.6 o .

Problem : Two sides of a triangle and an angle opposite one of them is given. There is no solution to the triangle. What must be true of the side opposite the given angle and the other given side?

The side opposite the given angle is shorter than or equal in length to the other given side. If it were longer, a solution would exist.

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