At a glance. One lesson of about 60 minutes. It assumes the unit circle (lesson 1), right-angled trigonometry and the angle sum of a triangle. Students derive the rule from two right-angled triangles, use it for a side and an angle, and decide how many triangles fit given two sides and a non-included angle. The cosine rule is the next lesson.
Calculator: scientific calculator. A scientific calculator is needed for sines and inverse sines. Degree mode and brackets are the two practical points; the figures give values to check against.
What students will learn
Know
- The sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C.
- The angle must be opposite the side in each pair.
- sin (180° − x) = sin x.
Understand
- The rule comes from finding the height of a triangle two ways.
- Two sides and a non-included angle can fit two triangles.
- If the given side opposite the angle is at least as long as the other given side, only one fits.
Do
- Find a missing side with the sine rule.
- Find a missing angle, and test the second possibility.
- Decide how many triangles fit.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the rule, opposite pairs and the second angle (check and questions 1 to 8).
- Criterion B. Investigating patterns: the derivation in the first figure and the compass arm in the second. Ask students to explain why the height is the same.
- Criterion C. Communicating: labelling the triangle, writing the rule and testing the second angle in words.
- Criterion D. Applying mathematics in real-life contexts: distances across a river or between towers, where no right angle can be placed.
The one thing to do with the figure
Make students commit before the height appears. In the first figure, ask “Which two expressions equal the height h?” before the readouts are read.
In the second figure, ask “How many triangles for a = 7?” before moving the slider, and ask what the smallest a is that works.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the smoke on the ridge. Collect what students think they need. |
| 10 min | First figure and key idea: the derivation of the rule. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose B, ask what is left for angle C when B = 130°. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 40°; 55°; 10. |
|---|---|
| Hook | Two angles and the 5 km side fix the triangle; then the third angle follows and the sine rule gives both distances. Two sides and a non-included angle may fit two triangles. |
| Worked examples | b = 17.5; B = 55.7° or 124.3°. |
| Faded example | Q = 33.0°, with only one triangle. |
| 1. How many triangles | C. A would need a shorter than the height; B forgets that the angles must add to less than 180°; D ignores that three measurements are given. |
| 2. Sine rule for a side | 20.6. 14 × 0.9063 ÷ 0.6157. |
| 3. Sine rule for an angle | 43.6. sin⁻¹ 0.689. |
| 4. The ambiguous case | 131.4. 180 − 48.6. |
| 5. When a triangle exists | 12. 25 × sin 28° = 11.7. |
| 6. A surveying problem | 103.4. 80 × sin 71° ÷ sin 47°. |
| 7. Sine rule with an obtuse angle | 21.3. 30 × sin 40° ÷ sin 115°. |
| 8. Reading the ambiguous case | B. A misses the second angle; C is the complement; D leaves out sin 32°. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 (the check) | Thinks the side cannot reach the base, keeps 130° without testing the angle sum, or thinks more is needed. | Ask: what is left for angle C if B = 130°? |
| 9.5, 23.9 or 12.7 in question 2 | Swaps the sines, uses angles, or forgets to divide. | Write the pairs p, P and q, Q and then the equation. |
| 0.69, 136.4 or 51.3 in question 3 | Stops at the sine, gives the obtuse angle, or uses angles. | Finish with the inverse key and then test. |
| 48.6, 41.4 or 101.4 in question 4 | Gives the first angle, the complement, or the third angle. | Ask: what other angle has the same sine? |
| 11, 25 or 22 in question 5 | Rounds down, uses b, or uses cosine. | Sketch the triangle and mark the perpendicular height. |
| 61.9, 85.7 or 74.7 in question 6 | Swaps the sines, or pairs the side with the wrong angle. | Find C first and mark each opposite pair in colour. |
| 42.3, 14.0 or 19.3 in question 7 | Swaps the sines, finds b instead of c, or forgets to divide. | Find C from the angle sum first; say: sin 115° is the same as sin 65°. |
| A, C or D in question 8 | Stops at the calculator's angle, uses the complement, or drops a sine. | Find sin C, then both angles, then test each against 32°. |
Questions worth asking
- Why must the angle and the side be opposite each other? The rule says each side is proportional to the sine of the angle opposite it.
- Why can an obtuse angle be used with the sine rule? sin (180° − x) = sin x, so the rule holds whichever angle is obtuse.
- When is there only one triangle in the ambiguous case? When the side opposite the given angle is at least as long as the other given side.
- How does the compass arm show the number of triangles? The arc of length a meets the base 0, 1 or 2 times.
Differentiation
A way in for anyone stuck
- Give a labelled triangle template with sides a, b, c opposite angles A, B, C and the rule written beside it.
- Use a checklist: label; write the pair; write the equation; solve; test the second angle.
- Provide the figure's compass picture on paper and a pair of compasses for the ambiguous case.
Stretch
- Show that a ÷ sin A is the diameter of the circle through A, B and C, by finding it in a right-angled triangle inscribed in a semicircle.
- Find the range of values of a for which A = 35°, b = 10 gives two triangles, and say what happens at the ends.
Students learning in English as an additional language
“Opposite” means across from: side a is across from angle A. “Acute” means less than 90 degrees and “obtuse” means between 90 and 180 degrees. “Ambiguous” means it could be read in two ways. “Perpendicular” means at a right angle.
| Word | Plain meaning | Example |
|---|---|---|
| sine rule | a side divided by the sine of its opposite angle is the same for all three sides | a ÷ sin A = b ÷ sin B |
| opposite | across from: side a is opposite angle A | the side that does not touch the angle |
| ambiguous case | a set of measurements that fits two different triangles | two sides and an angle not between them |
| perpendicular height | the shortest distance from a corner to the opposite side | the height from C to AB |
| obtuse | an angle between 90° and 180° | an angle of 120° |
Sentence starters for explaining aloud or in writing:
- Side … is opposite angle …, so ….
- The calculator gives …, and the other angle is …, which does or does not fit because ….
- The side a is longer or shorter than b, so there are … triangles.
Two things not to say
“The sine rule only works in acute triangles.” It holds for every triangle.
“Always give two answers.” The second angle has to be tested.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Third angleTwo angles of a triangle are 72° and 51°. Find the third.
57°. - Sine rulea = 10, A = 30°, B = 90°. Find b.
20. 10 ÷ 0.5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo towers 4 km apart see a fire at angles 55° and 70° to the line between them. How far is the fire from the tower with the 55° angle?
4 × sin 70° ÷ sin 55° = 4.6 km. - Earlier work (Lesson 4)Solve sin x = 0.5 for 0° to 360°.
30° and 150°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA = 40°, a = 5 and b = 10. How many triangles? Explain.
None: the height is 10 sin 40° = 6.4, more than 5. - Working backwardsTwo triangles fit with A = 30° and b = 10. Give a value of a that works.
Any a between 5 and 10, for example 7.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Give each pair a pair of compasses and ask them to draw A = 35°, b = 10 and then swing arcs of different lengths a to see how many triangles close.