The Law of Sines

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Ambiguous Case of the Law of Sines

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Key Questions

  • Yes, it does. It does not have to be a a right triangle.

    I ope that this was helpful.

  • First of all it is useful to say the notation in a triangle:

    Opposite at the side #a# the angle is called #A#,
    Opposite at the side #b# the angle is called #B#,
    Opposite at the side #c# the angle is called #C#.

    So, the Sinus Law can be written:


    This Law is useful in all the cases SSA and NOT in the case SAS, in which the Law of Cosinus has to be used.

    E.G.: we know #a,b,A#, then:

    #sinB=sinA*b/a# and so #B# is known;

    #C=180°-A-B# and so #C# is known;


  • law of sines relates two sides and the angles opposite them. So
    any time you have two angles (and then can easily figure out the
    third), it's easy to use the law of sines: ASA or AAS.

    In the SSA case, you can use the law of sines, but you have to
    remember that when you get something like sin(alpha) = 0.7 or
    whatever, there might be two possible triangles. You have to check
    alpha = 44.427 degrees or so, but it's also possible that alpha =
    135.573 degrees or so.

    So in the SSA case, the law of sines is easier, but you have to
    remember to check for that second possibility i.e, law of cosines...

  • Sometimes when given two sides and a not-included angle, there are two possible triangles since there are two angles between 0 and 180 degrees that have the same sine value.

    When solving this kind of triangle, the first step is to use the Law of Sines to find the angle opposite the adjacent side. However, solving the Law of Sines gives you only the Sine of the angle, not the angle measure itself.
    Typically you would then use a calculator to evaluate the Arcsine of the resulting value. This will yield a value between 0 and 90 degrees. However, the Supplement of that angle, 180 minus its measure, has the same Sine value.
    After solving the triangle for the value given by the arcsine, find the supplement of that new angle, add it to the original given angle, and if the sum is less than 180 there will be a second triangle having that supplementary value. Use that supplementary value along with the givens to find the rest of the parts of the second triangle.


  • Zsigmond S. answered · 4 months ago