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Here is the problem: Two sides of a triangle are 3 and 8 with a 45° angle between them. Find the third side.

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SAS: find the third side and the remaining angles.

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Substitute into c squared equals a squared plus b squared minus 2ab times cos C.

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Take the square root.

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Find angle A with the law of cosines rearranged.

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Angles of a triangle add to 180°.

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Area from two sides and the included angle.

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So the answer is c approximately 6.24971; A approximately 19.841968°, B approximately 115.15803°.

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The triangle has sides a equals 3, b equals 8, c approximately 6.24971 and angles A approximately 19.841968°, B approximately 115.15803°, C approximately 45°. It is an obtuse scalene triangle with area approximately 8.4852814 and perimeter approximately 17.24971.

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A plus B plus C equals 180°.

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Heron prime s formula gives the same area: 8.4852814.

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Law of sines ratios: a/sin A approximately 8.8384246, b/sin B approximately 8.8384246, c/sin C approximately 8.8384246.