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Find side length using trig

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  1. Jun 29, 2015 #1
    1. The problem statement, all variables and given/known data

    $AB = 20 cm$, $m∠A = 30°$ , and $m∠C = 45°$ . Express the number of centimeters in the length of $BC$ in simplest radical form.

    2. Relevant equations
    $sin A = sin C$

    3. The attempt at a solution
    $AB = 20, BC = x$

    D is the point where this obtuse triangle separates into 2 right triangles

    $BD/20 = sin A$
    $AD/20 = cos A$

    30-60-90 triangle
    $1:2:\sqrt{3}$

    BD is 10 according to this ratio which means that sin A is 1/2 and AD would be $20\sqrt{3}$

    sin C is the same but for a 45-45-90 triangle instead.

    45-45-90 triangle
    $1:1:\sqrt{2}$

    But here is where I am stuck. I am trying to find the side lengths of the 45-45-90 triangle with the trigonometric ratios being the same for both triangles but the angles being different so that I know the hypotenuse BC. But I don't know what side lengths will give me the trigonometric ratios being the same and the $1:2:\sqrt{3}$ and $1:1:\sqrt{2}$ side length ratios being true.
     
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  3. Jun 29, 2015 #2

    SammyS

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    Your result for the length of side AD is incorrect.
     
  4. Jun 29, 2015 #3

    RUber

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    I second what SammyS says.
    Once you know BD though, you should quickly know BC, since BC is the hypotenuse of the 45-45-90 triangle, right? You have already written the appropriate ratio for the length of a side to the hypotenuse of this triangle.
     
  5. Jun 29, 2015 #4

    HallsofIvy

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    No, with A= 30 degrees and B= 45 degrees sin(A) is definitely not equal to sin(C)!
    Perhaps you meant the sine law:
    [tex]\frac{sin(A)}{BC}= \frac{sin(B)}{AC}= \frac{sin(C)}{AB}[/tex]
    The cosine law might also be useful:
    [tex](AB)^2= (AC)^2+ (BC)^2- 2(AC)(BC) cos(C)[/tex]
    and equivalent formulas for the other two angles.

     
  6. Jul 21, 2015 #5
    I'm kind of new here... can someone please tell me what the dollar signs represent?
     
  7. Jul 21, 2015 #6

    Mentallic

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    They should be double dollar signs, and they're just a simpler alternative to the [tex] tags.
     
  8. Jul 21, 2015 #7

    SammyS

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    To further explain:

    The dollar signs (as well as the # sign ) are used as tags to enable using "LaTeX" for displaying mathematical expressions.

    On this site, those should be doubled.

    For instance, $20\sqrt{3}$ , should have been ## $ \$ 20\text{\sqrt } 3 ## ## $$ ## .

    It would display $$20\sqrt{3}$$
     
    Last edited: Jul 21, 2015
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