Yes
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Yes, the law of sines can be used in a right triangle. The law applies to any arbitrary triangle.
Land surveying makes an extensive use of the sine and cosine law. The idea is to subdivide the land into many triangles and to measure one side and two angels of each triangle. With the sine law the other two sides can be computed. The Mount Everest was found by this method to be the highest mountain on planet earth.
The law of sines states the in a triangle with sides of length a, b, and c, and angles of A, B, and C where each angle's letter corresponds with the side opposite it, that Sin(A)/a = Sin(B)/b = Sin(C)/c. The law of cosines states that for the same triangle, c2 = a2 + b2 -2abCos(C).
The law of tangents can be used to determine another side length; the sine rule can then be used to determine the third side length.
Even though the cosine of an angle was not a known concept at the time, Euclid (3rd century BC) in Egypt stated and proved a pair of propositions which were equivalent to the law of cosines. One proposition was applicable for obtuse angles and the other to acute angles.