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

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

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

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 ≈ 0.959 and therefore B ≈ 73.6⁰ Author: Culan O'Meara

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(180-73.6) have same Sine values Author: Culan O'Meara

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 ≈ 106.4⁰ 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

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