Trigonometry

When you have a triangle that is not a right triangle, and let's say you are using the Law of Cosines to determine an angle measure, what side do you use in place of c in the Pythagorean Theorem portion of the Law? For example, if you have a triangle with 13, 10, and 9 (and not a right triangle to distinctly have a hypotenuse measure) do you use the longest side for the formula? Hoping this makes sense. The question I'm working on is multiple choice, and every option is an answer depending on the way you solve with which numbers. Thanks.

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  1. You can label it any old way.
    a^2 = b^2+c^2 - 2 b c cos A
    the trick is to use the cosine of the angle opposite the side you are solving for.
    If a = 13, b = 10, c = 9 then
    a^2 = b^2 + c^2 - 2 bc cos A
    b^2 = a^2 + c^2 - 2 ac cos B
    c^2 = a^2 + b^2 - 2 ab cos C

    Te last one is the one you see in your text, but any angle in the triangle can be C as long as you solve for the side opposite that angle.

  2. Just adding another thought.

    I usually find the angle opposite the largest side, just in case that angle is obtuse. Your calculator will give you that obtuse angle using the inverse cosine function.
    In most cases, once you find one of the angles using the cosine law, one of the other angles can be found using the sine law, and then the third angle can be found using the 180° sum principle.
    However, if you find a small angle using the cosine law, you could run into the ambiguous case, one of the other angles could be either obtuse acute (the sine is positive in quads I and II), but your calculator would give you only the acute angle.
    e.g. Suppose in triangle ABC, a=8, b=4 and c = 5
    Finding angle B:
    4^2 = 5^2 + 8^2 - 2(5)(8)cosB
    cosB = (25+64-16)/80
    B = 24.1°
    Now using the Sine Law:
    sinA/8 = sin24.1/4
    sinA = 8sin24.1/4 = .8166...
    A = 54.9°, which then makes C = 101° , but clearly A must be the largest angle since it is opposite the largest side.

    The explanation is that from
    sinA = 8sin24.1/4 = .8166...
    A could be 54.9° or 180-54.° which would be 125.2° , the correct answer.

    Finding the largest angle using the Cosine Law avoids that ambiguous case.

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