Common content, so this runs for both courses.
Set the mast question before any rule is named. Sides 5, 6 and 8; find the angle opposite the 8. Let them use whatever they like. A good proportion of the room will produce 87.1°, because the sine rule is the one they reach for, and they will be confident.
Then ask for the three angles and the total. 174.27°. The room finds its own error, which is worth more than being told, and the figure then explains it rather than announcing it.
Stop on the fourth view, the cosine curve. The question to ask there is "how many times does a horizontal line cut this one?" Once. That is the whole reason the cosine rule is safe and the sine rule is not.
| Question | Answer |
|---|---|
| 1. Third side, 7 and 9 with 52° | c² = 52.43, so c = 7.24. |
| 2. Largest angle of a 5, 6, 8 triangle | cos C = −3/60 = −0.05, so 92.9°. |
| 3. Angles totalling 174.27° | B. The sine rule returned the acute partner. |
| 4. Area, sides 5 and 6 about 92.9° | ½ × 5 × 6 × sin 92.9° = 14.98. |
1 markThe right rule, stated with the letters matched to the triangle.
1 markCorrect substitution, including the included angle where the formula requires one.
1 markThe answer to the stated accuracy.
In a multi-part question an obtuse angle found by the sine rule usually loses the final mark and then everything built on it. Examiners do carry follow-through, but only where the subsequent working is consistent, and a 174.3° angle sum often is not.
| They wrote | What happened |
|---|---|
| 87.1° | Sine rule on an obtuse angle. The commonest answer, and the point of the lesson. |
| 48.5° | Paired the wrong side with the angle. Worth dwelling on: 48.5° is a genuine angle of that triangle, so no check they own will catch it. |
| 14.41 | On question 1, added the 2ab cos C term. They have the formula and not the sign. |
| 52.43 | On question 1, stopped at c squared. |
| 12.27 | On question 1, in radians. cos 52 radians is −0.163. |
| 29.96 | On question 4, dropped the half. |
| 19.97 | On question 4, used 5 and 8, which do not straddle the angle. |
"Why can't I just always use the cosine rule?" For an angle from three sides, you can, and should. It will not start from two angles and a side, because it needs two sides and you have one. Let them try it and see where it stalls. Expect the follow-up that the third angle comes free from the angle sum, which is true, and is exactly why the sine rule is the right tool there.
"How am I supposed to know it was obtuse?" You are not, before you start. That is the argument for finding the largest angle first with the cosine rule: the sign of the cosine then tells you, and every angle left over is guaranteed acute.
"Is the ambiguous case on the exam?" No. Both guides exclude it from this sub-topic. But it is not an abstraction: this page is the ambiguity arriving through the back door, because the student chose the sine rule when they did not have to.
| Stage | What to do |
|---|---|
| Demonstrate | Type cos⁻¹((25+36−64)/(2×5×6)) in one go and get 92.9°. Then type it without the brackets round the bottom and let the machine throw its error, so they see that this particular slip announces itself. Then do the wrong pairing, 48.5°, and point out that this one does not. |
| Where they stick | Radians: cos⁻¹(−0.05) comes back as 1.62 and looks like a different sort of mistake. Also typing cos 52 and then squaring, rather than squaring inside. |
| The check | Before anyone presses a key, ask whether the cosine will be positive or negative. If they can say "negative, because 8² is bigger than 5² + 6²", 64 against 61, they understand the rule. Then the sign of what appears confirms or contradicts it. |
Ask which machine each student has before the lesson, not during it. Degrees mode is the setting that ruins this whole topic, and the two machines show it in different corners of the screen.
| Step | What |
|---|---|
| 1 | Set the mast question cold. Do not name a rule. Collect answers. |
| 2 | Ask for all three angles and the total. Let 174.27° land. |
| 3 | Run the figure. Stop on the cosine view and ask how many crossings. |
| 4 | Build the "which rule starts" table with them, from the five cases. |
| 5 | Cosine rule forwards on 7, 9 and 52°, then area, which is quick and gives the room a win. |
| 6 | Close on the habit: largest angle first, cosine rule. |
Do not say "remember to check for the obtuse case". It is a memory instruction, it will be forgotten under pressure, and the course does not set the ambiguous case anyway. The instruction that works is about which rule to pick, not about remembering a second answer.
Do not call the cosine rule the harder one. Students avoid it for that reason and walk into this. It is one substitution into one formula and it cannot return the wrong angle.