At a glance. One lesson of about 60 minutes. Naming opposite, adjacent and hypotenuse; sin, cos and tan as ratios; choosing a ratio; finding a missing side, with the unknown on top or underneath.
Calculator: scientific calculator. Values of sin, cos and tan of most angles are not nice numbers; degree mode is part of the skill.
What students will learn
Know
- sin θ = opposite ÷ hypotenuse.
- cos θ = adjacent ÷ hypotenuse.
- tan θ = opposite ÷ adjacent.
Understand
- The ratios depend only on the angle because right-angled triangles with the same angle are similar.
- The choice of ratio depends on which two sides are involved.
Do
- Label the sides relative to an angle.
- Choose the correct ratio.
- Find a missing side with the unknown on top or underneath.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: naming sides and applying the ratios (questions 1 to 7).
- Criterion B. Investigating patterns: with the figure, students find that a ratio is the same for any hypotenuse, that sin and cos swap values between 30 and 60 degrees, and that they are equal at 45 degrees.
- Criterion C. Communicating: sides are labelled on a sketch, the ratio is written, then rearranged.
- Criterion D. Applying mathematics in real-life contexts: ramps, trees and ladders (questions 4 and 6).
The one thing to do with the figure
Fix the angle at 30° and move the hypotenuse from 4 to 12. Ask which readouts change. The lengths move; the three ratios do not.
Then move the angle to 60° and ask what has happened to sin and cos. This is the first hint that sin θ = cos(90° − θ), which comes up again in later years.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the two ramps; vote on whether the height divided by the length is the same. |
| 12 min | The figure; naming sides and the three ratios. |
| 12 min | Worked examples; the 25 degree triangle in pairs. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 12 cm; the hypotenuse; x = 4. |
|---|---|
| Worked examples | Ramps: 1.03 m and 2.05 m. Ladder: 4.3 m. |
| Faded example | 4.7 cm. |
| 1. Naming the sides | A, BC. B is the adjacent side; C is the hypotenuse. |
| 2. Sine as a ratio | 0.6 = 6 ÷ 10. |
| 3. Tangent as a ratio | 0.417 = 5 ÷ 12. |
| 4. Missing side with sin | 6.9 m. |
| 5. Missing side with cos | 9.2 cm. |
| 6. Missing side with tan | 15.6 m. |
| 7. Missing hypotenuse | 14.9 cm. |
| 8. Perimeter | 23.3 cm. |
| 9. Choosing the ratio | B. A multiplies when the unknown is underneath; C uses sin, which goes with the opposite side. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 0.8 or 0.75 in question 2 | Picks the wrong ratio. | Ask which two sides sine uses. |
| 2.4 or 0.385 in question 3 | Divides the wrong way round, or uses the hypotenuse for tan. | Ask: is tan of a small angle bigger or smaller than 1? Which two sides does tan use? |
| 9.8, 8.4 in question 4; 11.8 in question 5 | Wrong ratio for the sides involved. | Label the sides first, then choose. |
| 24.4 in question 5, 25.6 in question 6 | Divides when the unknown is on top. | Ask: should the answer be bigger or smaller than the side you started with? |
| 12.3 in question 6 | Uses sin with the adjacent side. | Label the 20 m: is it opposite, adjacent or hypotenuse? |
| 3.3 or 13.2 in question 7 | Multiplies when the unknown is underneath, or uses tan for the hypotenuse. | Rearrange the equation sin 28° = 7 ÷ h before using it. |
| 13.3 or 19.1 in question 8 | Leaves a side out of the perimeter. | Ask them to list the three sides first. |
| A in question 9 | Multiplies when the unknown is underneath. | Ask: must the hypotenuse be longer or shorter than 9? |
Questions worth asking
- Why does sin 30° not depend on the size of the triangle? Listen for “all right-angled triangles with a 30 degree angle are similar”.
- Which ratio would you use with the opposite and the hypotenuse? Sin. Ask for a sentence in the form “I know … and want …”.
- Can sin ever be bigger than 1? No: the hypotenuse is the longest side.
Differentiation
A way in for anyone stuck
- Provide a labelled triangle template with opp, adj, hyp and a table of the three ratios.
- Give the choice rule as a table: know and want, ratio, equation.
- Let students use the figure to check their answers to Level 1 before the calculator.
Stretch
- Show that sin² θ + cos² θ = 1 for the 30 degree triangle (0.25 + 0.75).
- Explain why tan θ = sin θ ÷ cos θ.
Students learning in English as an additional language
Use the three-part word split on the board: op-po-site, ad-ja-cent, hy-po-te-nuse, with a colour for each side.
| Word | Plain meaning | Example |
|---|---|---|
| hypotenuse | the side opposite the right angle; the longest side | the ramp itself |
| opposite | the side across from the angle you are using | the height of the ramp |
| adjacent | the side next to the angle you are using, not the hypotenuse | the ground distance |
| sine | opposite divided by hypotenuse | sin 30° = 0.5 |
| cosine | adjacent divided by hypotenuse | cos 60° = 0.5 |
| tangent | opposite divided by adjacent | tan 45° = 1 |
| angle of elevation | the angle measured up from the horizontal to what you are looking at | the top of the tree in question 6 |
Sentence starters for explaining aloud or in writing:
- I know the … and I want the …, so I use …
- The side opposite θ is … because …
- My answer is sensible because the hypotenuse is the …
Two things not to say
“SOH CAH TOA, remember it.” A mnemonic without the ratio idea gives the wrong-way-round slips; use it after the figure, not instead.
“Always multiply.” The unknown underneath needs a division, and that is where the 24.4 and 3.3 slips come from.
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.
- RatioOpposite 3, hypotenuse 5. Find sin θ.
0.6. - SideHypotenuse 20, angle 30°. Find the opposite side.
10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- SlopeA path rises at 12° and is 50 m long. How high does it rise to 1 d.p.?
10.4 m. - ShadowThe sun is at 50° to the horizontal. A pole 6 m tall casts a shadow of what length (1 d.p.)?
5.0 m.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningtan 45° = 1. What does that say about the opposite and adjacent sides?
They are equal. - From Lesson 5A triangle with a 30° angle and hypotenuse 8 is enlarged by scale factor 3. What is sin 30° for the new one?
Still 0.5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: a protractor and ruler can draw 20 degree triangles of different sizes, and measuring the ratio is a good extension.