## anonymous 4 years ago Law of Sines- The Ambiguous Case 1) m<C=29 degrees b=31 m c=27 m

1. anonymous

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2. anonymous

I have to find m<A

3. anonymous

I don't get how to start. :/

4. anonymous

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 $B=\sin^{-1}(.5566)=33.82$

5. anonymous

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6. anonymous

sorry for the lousy picture, but you can see you have two choices for angle A

7. anonymous

in big triangle angle A is 180-29-33.82 = 117.18

8. anonymous

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.

9. anonymous

these are not right triangles so you do not have "opposite" and "adjacent"

10. anonymous

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

11. anonymous

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:

12. anonymous

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13. anonymous

How is this determined? opp $\ge$ adj or opp $\le$ adj

14. anonymous

that is the larger triangle, and first we solve for B because you have angle C, side c, and side b

15. anonymous

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

16. anonymous

i labled them with the side c opposite angle C, side a opposite angle A and side b opposite angle B

17. anonymous

then you can use $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$[

18. anonymous

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

19. anonymous

So the answer to find m<A is 117 degrees?

20. anonymous

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

21. anonymous

*has angle A = 117.18

22. anonymous

the smaller triangle has a different angle, but that is now easy to find

23. anonymous

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24. anonymous

Thank you so much! I got the hang of this now :)

25. anonymous

the other angle B, the one in the smaller triangle, has measure 180 - 33.82 = 146.18

26. anonymous

so the measure of angle A in the smaller triangle is $180 - 29 - 146.18 = 4.82$