The law of cosines with a right angle is just the pythagorean theorem. The cosine of 90 degrees is 0. That is why the hypotenuse squared is equal to the sum of both of the legs squared
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Having sufficient angles or sides one can use either, The Law of Sines, or, The Law of Cosines. Google them.
Yes
Either.However, if you know two sides and the includedangle then the sine rule is simpler.
If you have two other angles, then add up those 2 and subtract that from 180. if you have all 3 sides then use the law of cosines: a squared = b squared + c squared - 2bc (cos A) If you have one angle and the 2 included sides, use the law of cosines as well. if you have an angle and the length of its opposite side, and the side opposite to the angle you want, then use the law of sines: sin A/ a = sin B/ b if you have the angle and the length of its opposite side and another angle, use the law of sines to figure out the unwanted angle anyway and then follow situation 1.
a2+b2=c2, but that only works for right triangles, where c is opposite the right angle. The law of cosines, see Related Link below, will help for non right triangles, but you need to know one of the angles.