791 8.2 The Law of Cosines Four possible cases can occur when we solve an oblique triangle. They are summarized in the following table. In all four cases, it is assumed that the given information actually produces a triangle. Four Cases for Solving Oblique Triangles Oblique Triangle Suggested Procedure for Solving Case 1: One side and two angles are known. (SAA or ASA) Step 1 Find the remaining angle using the angle sum formula 1A + B + C = 180°2. Step 2 Find the remaining sides using the law of sines. Case 2: Two sides and one angle (not included between the two sides) are known. (SSA) This is the ambiguous case. There may be no triangle, one triangle, or two triangles. Step 1 Find an angle using the law of sines. Step 2 Find the remaining angle using the angle sum formula. Step 3 Find the remaining side using the law of sines. If two triangles exist, repeat Steps 2 and 3. Case 3: Two sides and the included angle are known. (SAS) Step 1 Find the remaining side using the law of cosines. Step 2 Find the smaller of the two remaining angles using the law of sines. Step 3 Find the remaining angle using the angle sum formula. Case 4: Three sides are known. (SSS) Step 1 Find the largest angle using the law of cosines. Step 2 Find either remaining angle using the law of sines. Step 3 Find the remaining angle using the angle sum formula. Heron of Alexandria (c. 62 CE) Heron (also called Hero), a Greek geometer and inventor, produced writings that contain knowledge of the mathematics and engineering of Babylonia, ancient Egypt, and the Greco-Roman world. Heron’s Formula for the Area of a Triangle A formula for finding the area of a triangle given the lengths of the three sides, known as Heron’s formula, is named after the Greek mathematician Heron of Alexandria. It is found in his work Metrica. Heron’s formula can be used for the case SSS. Heron’s Area Formula (SSS) If a triangle has sides of lengths a, b, and c, with semiperimeter s = 1 2 1 a +b +c2, then the area of the triangle is given by the following formula. =!s1s −a2 1s −b2 1s −c2 That is, the area of a triangle is the square root of the product of four factors: (1) the semiperimeter, (2) the semiperimeter minus the first side, (3) the semiperimeter minus the second side, and (4) the semiperimeter minus the third side.
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