14)+Given+a+triangle+with+the+ambiguous+case+of+law+of+sines,+determine+the+number+of+possible+triangles+then+solve

By Brooke Ransom

Why Ambiguous Case? I chose to do my project on ambiguous cases because they are one of the more difficult concepts to grasp and they are not put in front of you in a problem; you have to find them. During the section on law of sines these cases were discussed but I do not believe many people understood what the ambiguous case was, how to find it, and how it effects answers in the problem. But the ambiguous case is special because you may not know it exists until the end of a problem where your answer makes no sense and you have to know how to go back and look for it to get the right answer. Ambiguous case can effect many problems and it reoccurs in other sections so it is an important concept to understand.

What is Ambiguous case? This can occur whenever you are given the Angle, Side, Side measurements of a triangle and use the law of sines.

View Link Below for Explanation of Ambiguous Case

[|SCAN0052.JPG]

EXAMPLE 1:



__sin(30)__ = __sin(C)__ 7 16

16sin(30) = 8 8/7 = 1.1429 sin(C) cannot be greater than or equal to 1 so given these dimensions the triangle cannot exist

Also because 16sin(30) = 8 and a = 7 you know that no triangle can exist because of the ambiguous case rule that when a < bsinA no triangle is possible, in this triangle c is equivalent to b.

EXAMPLE 2:

__sin(30)__ = __sin(C)__ 10 16

16sin(30) = 8 8/10 = .8 sin(C) = .8 C = 53 degrees therefore B = 97 degrees



bsinA = 16sin(30) = 8 8 < 10 < 16 bsinA < a < b, so we know __2__ triangles are formed

To find the second triangle you subtract angle C from 180 degrees to find the other possible angle formed

180 - 53 = 127 degrees = C therefore B = 23 degrees



Two Triangles are Possible!!! (I apologize for the weird set-up of fractions in the Law of Sine calculations, I can`t get them correct, but to clarify the first number goes under the first sine and the second number should be farther right underneath the second sine)

Instructional Video:

[]

Problem Set: 1) Solve the triangle given c = 13 ft, b = 23 ft, and C = 32 degrees 2) How many triangle are formed given a = 2 in, b = 1.5 in, and A = 40 degrees

Website used: ---http://www.regentsprep.org/Regents/math/algtrig/ATT12/lawofsinesAmbiguous.htm

---http://www.google.com/imgres?q=ambiguous+case+triangle&um=1&hl=en&sa=N&biw=1440&bih=719&tbm=isch&tbnid=FWH18tf2KXofHM:&imgrefurl=http://www.sparknotes.com/math/trigonometry/solvingobliquetriangles/section3.rhtml&docid=gPC4uD5JJ_Eg8M&imgurl=http://img.sparknotes.com/figures/E/e60eedf4d6214df383cf2203d899e707/obtusegiven.gif&w=475&h=350&ei=A0N-T7urLYOu8QT8ydyfDg&zoom=1&iact=hc&vpx=383&vpy=164&dur=130&hovh=193&hovw=262&tx=127&ty=86&sig=116315772825402880492&page=1&tbnh=163&tbnw=228&start=0&ndsp=16&ved=1t:429,r:1,s:0,i:71//

//---http://www.google.com/imgres?q=ambiguous+case+triangle&um=1&hl=en&sa=N&biw=1440&bih=719&tbm=isch&tbnid=8R1d0EpzKGbP2M:&imgrefurl=http://www.sparknotes.com/math/trigonometry/solvingobliquetriangles/section3.rhtml&docid=gPC4uD5JJ_Eg8M&imgurl=http://img.sparknotes.com/figures/E/e60eedf4d6214df383cf2203d899e707/acutegiven.gif&w=400&h=420&ei=A0N-T7urLYOu8QT8ydyfDg&zoom=1&iact=hc&vpx=202&vpy=154&dur=495&hovh=197&hovw=187&tx=88&ty=84&sig=116315772825402880492&page=1&tbnh=163&tbnw=155&start=0&ndsp=16&ved=1t:429,r:0,s:0,i:69

---http://www.themathpage.com/atrig/law-of-sines.htm

---http://www.mathwarehouse.com/trigonometry/law-of-sines/ambiguous-case-of-law-of-sines.php