At a glance. One lesson of about 60 minutes. It assumes sine, cosine and tangent in right-angled triangles from Year 9 and Pythagoras. Students use the sine rule and the cosine rule to find a side, the area formula, and decide which rule fits the given information. Finding angles, and the ambiguous case, are left for the Diploma.
Calculator: scientific calculator. A scientific calculator is needed for sines and cosines. The degree-mode check and bracketing the top of a fraction are the two practical points; the figures give the values to check against.
What students will learn
Know
- Side a is opposite angle A.
- The sine rule pairs each side with the sine of its opposite angle.
- The cosine rule needs two sides and the angle between them.
Understand
- The sine rule includes the right-angled triangle, where sin 90° = 1.
- The cosine rule includes Pythagoras, where cos 90° = 0.
- An obtuse angle makes the third side longer than Pythagoras gives.
Do
- Find a missing side with the sine rule.
- Find a missing side with the cosine rule.
- Choose a rule from the given measurements.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the rules, the opposite pairs, the area formula (questions 1 to 7).
- Criterion B. Investigating patterns: the first figure. Ask students to predict what the ratio will be for a given A and B before moving the sliders.
- Criterion C. Communicating: writing the rule used and the substitution, with units where there are any.
- Criterion D. Applying mathematics in real-life contexts: the lake, a surveyor who cannot cross the water, a triangle with no right angle.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “What happens to b ÷ sin B when A becomes 30°?” and take values before moving the slider.
In the second figure, ask what C = 90° will give, and then what a larger angle will give for the third side. Many students expect it to be shorter.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: across the lake. Collect what students think they need to measure. |
| 10 min | First figure and key idea: the sine rule and the cosine rule. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask where PQ and QR meet. |
| 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 | 5; 60°; 10. |
|---|---|
| Hook | She needs the length AC and the angles at A and at C: two angles and a side. Then the third angle B follows, and the sine rule gives AB. |
| Worked example | b = 8 × sin 70° ÷ sin 50° = 9.8; c2 = 94.96 so c = 9.7. |
| Faded example | 42 × sin 35° = 24.1 square units. |
| 1. Choosing a rule | B. A has no opposite pair; C needs a right angle; D finds area. |
| 2. Sine rule for a side | 8.5. 6 × 0.7071 ÷ 0.5. |
| 3. A side over the sine of its angle | 14. 7 ÷ 0.5. |
| 4. Cosine rule for a side | 6.1. √37.58. |
| 5. Sine rule with an opposite pair | 7.0. 12 × 0.5736 ÷ 0.9848. |
| 6. Area from two sides and an angle | 40.5. 63 × 0.6428. |
| 7. Cosine rule with an obtuse angle | 10.4. √109. |
| 8. Setting up the sine rule | A. B has the sines swapped; C is side b; D pairs 10 with the wrong angle. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Looks for an opposite pair, uses Pythagoras, or finds the area. | Ask: where is the angle, between the two sides or opposite one of them? |
| 4.2, 9 or 12 in question 2 | Swaps the sines, uses the angles themselves, or leaves out a sine. | Write the pairs a, A and b, B and then the equation. |
| 3.5, 0.07 or 0.23 in question 3 | Multiplies, inverts, or uses the angle instead of its sine. | Say: side over sine of the opposite angle. |
| 9.4, 37.6 or 8 in question 4 | Uses Pythagoras, forgets the square root, or leaves out the 2. | Write c squared on its own line before the root. |
| 20.6, 6.9 or 5.3 in question 5 | Swaps the sines, forgets to divide, or uses angles. | Mark the opposite pairs b, B and c, C with matching colours. |
| 81, 63 or 48.3 in question 6 | Leaves out the half, leaves out the sine, or uses the cosine. | Say the formula aloud: half, a, b, sine. |
| 6.2, 109 or 8.6 in question 7 | Subtracts instead of adding, forgets the root, or uses Pythagoras. | Ask for the sign of cos 120° first. |
| B, C or D in question 8 | Swaps the sines, finds side b, or pairs the side with the wrong angle. | Find C first, then write the pair c with C. |
Questions worth asking
- Why does the sine rule need an opposite pair? The ratio of a side to the sine of the angle opposite it is the same for all three sides, so one known pair gives the value.
- Why is Pythagoras a special case of the cosine rule? cos 90° = 0, so the correction term is zero.
- Why does an angle bigger than 90° give a longer third side? cos is negative there, so the correction term is added.
- How do you choose between the rules? An angle and its opposite side known: sine. Two sides and the angle between: cosine.
Differentiation
A way in for anyone stuck
- Give a labelled triangle template with a, b, c opposite A, B, C and the rule written beside it.
- Use a checklist: label the sides and angles; write the pair; write the equation; solve.
- Provide the calculator checks: sin 30 = 0.5 and cos 60 = 0.5 in degree mode.
Stretch
- Use the cosine rule to find angle C in a triangle with a = 5, b = 6 and c = 7. Write the rearranged rule first.
- Show that the area formula gives half the product of two perpendicular sides when C = 90°.
Students learning in English as an additional language
“Opposite” means across from: side a is across from angle A. “Included angle” means the angle between two sides. “Third side” is the one you want to find. “Obtuse” means bigger than 90 degrees.
| 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 |
| cosine rule | the square of a side equals the sum of the squares of the others minus 2ab cos C | used with two sides and the angle between them |
| opposite | across from: side a is opposite angle A | the side that does not touch the angle |
| included angle | the angle between two given sides | the angle at C between a and b |
| obtuse | an angle between 90° and 180° | an angle of 120° |
| degree mode | the calculator setting that reads angles in degrees | needed for sin 30 = 0.5 |
Sentence starters for explaining aloud or in writing:
- Side … is opposite angle …, so I use the … rule.
- I know two sides and the angle between them, so ….
- The third angle is 180° minus ….
Two things not to say
“Use sin, cos or tan, whichever you like.” The rule depends on what is known.
“The cosine rule is Pythagoras with extra bits.” Pythagoras is the special case when the angle is 90°.
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 84° and 37°. Find the third.
59°. - 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 wires 5 m and 8 m long are fixed to the top of a mast with 35° between them. How far apart are their lower ends?
c2 = 89 − 80 cos 35° = 23.5, so 4.8 m. - Earlier work (Lesson 4)The second differences of a table are all 6. Find a in ax2 + bx + c.
3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan a triangle have sides 3, 4 and an included angle of 90° and a third side of 6? Explain.
No: Pythagoras gives 5, so the third side is 5. - Working backwardsTwo sides are 6 and 8 and the third is 10. What is the included angle, and why?
90°, because 36 + 64 = 100, so cos C = 0.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All sliders 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 protractor, a ruler and a drawn triangle; ask them to measure the sides and compare a ÷ sin A with b ÷ sin B.