Law of Sines- The Ambiguous Case 1) m

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Law of Sines- The Ambiguous Case 1) m

Mathematics
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|dw:1328495762650:dw|
I have to find m
I don't get how to start. :/

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lets find the big triangle first. by the law of sines you have \[\frac{\sin(29)}{27}=\frac{\sin(B)}{31}\] \[\frac{31\sin(29)}{27}=\sin(B)\] \[\sin(B)=.5566\] rounded so \[B=\sin^{-1}(.5566)=33.82\]
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sorry for the lousy picture, but you can see you have two choices for angle A
in big triangle angle A is 180-29-33.82 = 117.18
That first response was great definitely helped! But what number would be the opposite & what number would be the adj? Cause that will help me determine the # of triangles.
these are not right triangles so you do not have "opposite" and "adjacent"
i was looking at the largter triangle first and using the laws of sines, because the invese sine funcion on your calculator will not give you the bigger angle, only the one less that 90 degrees
My teacher gave us notes and it says if you have two sides & one opposite (NON-included) angle of a triangle then there are three possibilities:
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How is this determined? opp \[\ge\] adj or opp \[\le\] adj
that is the larger triangle, and first we solve for B because you have angle C, side c, and side b
so you know three out of the 4 numbers for the ratio it is not a matter of "opposite" "adjacent" etc because these are not right triangles
i labled them with the side c opposite angle C, side a opposite angle A and side b opposite angle B
then you can use \[\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\][
and since we know side c, angle C, side b we have three out of the 4 numbers for the ratio here \[\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\] allowing us to solve for angle B
So the answer to find m
the reason this is the "ambiguous case" is that there are two possible triangles, the ones i drew in the first picture. in the bigger triangle and A is 117.18
*has angle A = 117.18
the smaller triangle has a different angle, but that is now easy to find
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Thank you so much! I got the hang of this now :)
the other angle B, the one in the smaller triangle, has measure 180 - 33.82 = 146.18
so the measure of angle A in the smaller triangle is \[180 - 29 - 146.18 = 4.82\]

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