1. LCHL Strand 3: Trigonometry
Sine Rule – The Ambiguous Case
Culan O’Meara – Ballinrobe Community School

2. Unit Circle and The Ambiguous Case
The word Ambiguous in this context means to be open to or having two possible answers
If you observe the Unit Circle here, you can see that Sine is positive in both the 1st(between 0⁰ and 90⁰) and 2nd Quadrant (between 90⁰ and 180⁰)
For Cosine, it is not possible to have more than one answer as Cosine is negative in 2nd Quadrant
Author: Culan O'Meara

3. Sine Rule - The Ambiguous Case
The three angles of a triangle add to 180⁰ so it’s possible, when given two sides and an angle to draw two triangles that fit the data
You can have a triangle with three acute angles and also a triangle that has one obtuse angle and two acute angles
Author: Culan O'Meara

4. Sine Rule - The Ambiguous Case
In the example shown, you are given a triangle with sides of length 6 and 8 and an angle of 46⁰ and asked to find the values of the missing angles and sides.
You can see that the triangle drawn contains 3 acute angles
If you solve this using Sine Rule, you will find that Sin B ≈ and therefore B ≈ 73.6⁰
Author: Culan O'Meara

5. Sine Rule - The Ambiguous Case
However, if we look back at our Unit Circle you will see that there are two angles that will give same answer(same reference angles and Sine is positive in both quadrants)
Both 73.6⁰ and 106.4⁰( ) have same Sine values
Author: Culan O'Meara

6. Sine Rule - The Ambiguous Case
In the triangle shown, we have drawn a different triangle with sides of length 6 and 8 and an angle of 46⁰
You can see that this triangle has B ≈ ⁰
Sin 106.4⁰ = Sin 73.6⁰ = 0.959
So having drawn two separate triangles, we finish off the question by finding the remaining angle and side in each case
Author: Culan O'Meara

7. Sine Rule - The Ambiguous Case
Author: Culan O'Meara