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Here is the problem: Two sides of a triangle are 10 and 12 with a 75° 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 13.486416; A approximately 45.743376°, B approximately 59.256624°.

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The triangle has sides a equals 10, b equals 12, c approximately 13.486416 and angles A approximately 45.743376°, B approximately 59.256624°, C approximately 75°. It is an acute scalene triangle with area approximately 57.95555 and perimeter approximately 35.486416.

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A plus B plus C equals 180°.

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Heron prime s formula gives the same area: 57.95555.

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Law of sines ratios: a/sin A approximately 13.962166, b/sin B approximately 13.962166, c/sin C approximately 13.962166.