1. Note 8– Sine Rule The “Ambiguous Case”
Note 8– Sine Rule The “Ambiguous Case”. Draw two different shaped triangles ABC in which c = 14m, a = 10m and A = 32. Hence find the size(s) of angle C.
This process (drawing triangles from verbal data and no diagram) takes time and practice. You need to access these types of problems and practise them thoroughly. Below is one possible diagram:
Now extend side AC1 past C1 to the new point C2 where the new length BC2 is the same as it was previously (10m)….. A B
14m
32 C1
10m A B
14m
32 C1
10m
The new ABC2 has the same given properties as the original ABC1 . Both triangles have c = 14, a = 10 and A = 32 . But note the angles at C are different! One is acute and the other obtuse.
10m C2

2. How are the two C angles related? (if at all)B
14m
32 C1
10m
TRIANGLE 1
TRIANGLE 2 B
14m
32 C1 C2
10m A
ANGLE C is obtuse
ANGLE C is acute
How are the two C angles related? (if at all)
Let angle BC2C1 = . B
14m
32 C1
10m C2
 angle BC1C2 = . (isos )
 angle BC1A = 180 –  (straight line)
10m
180 –    A
Conclusion: The (green) acute angle at C2 and the (blue) obtuse angle at C1 are supplementary. Thus, for example if one solution is 73 then the other solution is 180 – 73 = 107

3. Ans:  = 47.9 or 132.1 Back to the question!
Draw the triangle with the acute, rather than the obtuse, angle at C. B
14m
32 C2
10m A 
Applying the Sine Rule,
One solution (the acute angle which is the only one given by the calculator) is therefore 47.9 and the second solution (the obtuse angle) is 180 – 47.9 = 132.1
Ans:  = 47.9 or 132.1

4. The Sine Rule - Summary
The Sine Rule can be used to find unknown sides or angles in triangles.
The Sine Rule formula is
To use the Sine Rule, you must have
A matching angle and opposite side pair (two givens)
A third given and an unknown, which also make an angle and opposite side pair
When asked to find the size of an ANGLE, first check whether the problem could involve the ambiguous case (see Example 5). In that case, the two answers are supplementary – i.e. add to 180
When confronted with a problem where you have to decide whether to use the Sine Rule or the Cosine Rule, always try for the Sine Rule first, as it is easier.
In every triangle, the largest side is always opposite the largest angle. The side lengths are in the ratio of the sines of their opposite angles.

5. The Sine Rule - Summary In every triangle,
The largest side is always opposite the largest angle.
The middle sized side is always opposite the middle sized angle, and
The smallest side is always opposite the smallest angle
The ratio of any two side lengths is always equal to the ratio of the sines of their respective opposite angles. A b c
These are just re-shaped versions of the original sine rule formulae. B C a

6. Read other examples for Obtuse Angles on Page 245 –’The Ambiguous Case’
Then Exercise 12D.2