I´ve just completed a unit of work with a high ability year 9 set on trigonometry. We derived the sine rule (sinA/a = sinB/b), cosine rule(a² = b² + c² – 2bc.cosA) and area formula (A = 1/2absinC) using an ABC triangle. Having experienced misconceptions with these formulae before, I asked them to re-write the cosine rule by labelling the vertices differently.
a² = b² + c² – 2bc.cosA
b² = a² + c² – 2ac.cosB
c² = a² + b² – 2ab.cosC
Maybe I shouldn’t have derived them all with an ABC triangle but then the top formula does appear in their IGCSE formula book. Secondly, I thought that by getting them to identify either the sine or cosine rule using triangles which are labelled differently (See PowerPoint) that they would gain a deeper understanding of the situations in which the formulae should be used.
Unfortunately, after making a concerted effort to avoid this, I still found that a few of the students (I think 3 out of 21) got confused when trying to apply the ABC representation of the cosine rule to triangles labelled differently. E.g PQR triangle – see below.
I was struggling with how to make it clearer for them until I realised that they had to investigate this for themselves. I therefore decided to set a homework in which the main goal was to make a set of criteria to understand when to use the cosine rule and when to use the sine rule.
This is the Trigonometry Homework I set. Very specific criteria was built up by the students through taking the best bits of each groups criteria.
Non-Right Angled Trigonometry Criteria
In general, you can find a missing length or angle on a non-right angled triangle using the sine or cosine rule if you have 3 pieces of information. The one exception to this if if you have 3 angles and want to find a missing length.
1. Check to see if you can use the sine rule. It is possible to use the sine rule when:
a. You have 2 lengths and one angle and you need to find an angle. Rearranging the equation can be made simpler by using sinA/a = sinB/b.
b. You have 2 angles and one length and you need to find a length. Rearranging the equation can be made simpler by using a/sinA = b/sinB.
It is important to note that the lengths must be opposite the angles to use the sine rule. For example,
2. If you can not use the sine rule, try to use the cosine rule. It is possible to use the cosine rule when:
a. You have all 3 sides and need to find a missing angle. (cosA = (b² + c² – a²)/2bc)
b. You have 2 sides and the angle between them and you need to find a missing length. (a² = b² + c² – 2bc.cosA). The angle you use must be opposite to the missing length. Warning: Take care to square root your answer at the end!
3. If you cannot use either of the sine or cosine rule to find a missing length or angle, you may need to find a different length or angle first. For example,
4. There are occasions when you could use either rule. For example,
I hope that because the students devised this criteria themselves it will not become a list that they eventually try to memorise.
If you have any suggestions for the list either by improving a section or adding an important section, I and my students would be greatful.