At a glance. One lesson of about 60 minutes. Inverse sin, cos and tan; angles of elevation and depression; mixed side and angle problems.
Calculator: scientific calculator. Inverse ratios have no mental route: degree mode and the shift or second-function key are part of the skill.
What students will learn
Know
- The inverse ratios sin⁻¹, cos⁻¹ and tan⁻¹ give an angle from a ratio.
- Angles of elevation and depression are measured from the horizontal.
- The choice of ratio depends on which two sides are known.
Understand
- Each of sin, cos and tan gives the same angle for the same triangle.
- A bigger ratio of opposite to adjacent means a bigger angle.
Do
- Find an angle from two sides.
- Draw a right-angled triangle from an elevation or depression problem.
- Combine side-finding and angle-finding.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the inverse ratios (questions 2 to 4).
- Criterion B. Investigating patterns: with the figure, students predict the angle from the ratio, and find that all three ratios agree.
- Criterion C. Communicating: sides are labelled, the ratio is written before the inverse is used, and the angle carries its unit.
- Criterion D. Applying mathematics in real-life contexts: stairs, ladders, towers and cliffs (questions 4 to 8).
The one thing to do with the figure
Set opposite 3 and adjacent 4, and ask for a prediction of the angle. Most will say “about 30” or “about 40”; the figure says 36.9. Move opposite up to 4 and ask what happens to the angle.
Ask why sin, cos and tan all give the same angle. The answer is that they are three different ratios of the same triangle: a student who says “it is one triangle” has understood.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the ramp; guess more or less than 10 degrees. |
| 12 min | The figure; the inverse key and degree mode. |
| 12 min | Worked examples; the rectangle 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 | 5 cm; 0.75; taking the square root. |
|---|---|
| Worked examples | Ramp: 4.8°. Kite: 38.7°. |
| Faded example | 24.0°. |
| 1. Choosing the inverse ratio | B. A and C use a ratio that needs the side that is not given. |
| 2. Inverse cos | 53.1°. |
| 3. Inverse tan | 37.9°. |
| 4. Ladder angle | 70.5°. |
| 5. Elevation | 28.1 m. |
| 6. Slope of a stair | 33.7°. |
| 7. Hidden right angle | 66.4°. |
| 8. Depression | 92.3 m. |
| 9. Choosing the expression | C. D is the upside-down ratio; A and B use a ratio that needs the hypotenuse. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 36.9 in question 2 | Uses the wrong inverse. | Ask which two sides cosine uses. |
| 52.1 or 56.3 in questions 3 and 6 | Divides the wrong way round. | Ask: should tan of a small angle be small or large? |
| 0.78 or 0.7 | Stops at the ratio and forgets the inverse. | Ask: is 0.78 an angle in degrees? |
| 19.5 or 71.6 in question 4 | Wrong ratio for the sides given. | Label the sides before choosing. |
| 9.7 or 97.1 in question 8 | Multiplies or uses the hypotenuse. | Draw the triangle with the 18 degrees at the boat. |
Questions worth asking
- Why do sin, cos and tan give the same angle for the same triangle? Listen for “it is the same triangle, only the ratio changes”.
- Which ratio would you use if you only knew the opposite and adjacent? Tan.
- How does the angle of depression from the cliff relate to the angle at the boat? They are equal alternate angles (the horizontal lines are parallel).
Differentiation
A way in for anyone stuck
- Give a triangle template and a three-row table: sides known, ratio, equation.
- Provide the calculator key sequence on a card for inverse functions.
- Use whole-number ratios (0.5, 0.75) before 0.778.
Stretch
- A triangle has sides 5, 12 and 13. Find its two acute angles (22.6° and 67.4°) and check they add to 90°.
- Why can sin−1(1.2) not be found?
Students learning in English as an additional language
Use the words “up from the level” for elevation and “down from the level” for depression, with arrows.
| Word | Plain meaning | Example |
|---|---|---|
| inverse | the operation that undoes another | tan−1 undoes tan |
| angle of elevation | the angle measured up from the horizontal | looking up at a tower |
| angle of depression | the angle measured down from the horizontal | looking down at a boat |
| horizontal | level, like the surface of still water | the sea |
| rounding | giving a number to a stated accuracy | 37.87 is 37.9 to 1 decimal place |
Sentence starters for explaining aloud or in writing:
- I know the … and the …, so I use …
- The ratio is … and the angle is …
- The angle of elevation is measured from the …
Two things not to say
“Press shift then tan and hope.” Without a ratio written first, students use the wrong inverse.
“The angle is the answer to the division.” The division gives a ratio; the inverse key gives the angle.
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.
- Angletan θ = 1. Find θ.
45°. - AngleOpposite 3, hypotenuse 6. Find the angle.
30°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- SlopeA road rises 1 m for every 10 m along the horizontal. What angle is that to 1 d.p.?
5.7°. - LighthouseFrom a boat 120 m out, the top of a 50 m lighthouse has an angle of elevation of what (1 d.p.)?
22.6°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo angles in a right-angled triangle are θ and 90° − θ. If sin θ = 0.6, what is cos(90° − θ)?
0.6. - From Lesson 6Hypotenuse 12, angle 35°. Find the opposite side to 1 d.p.
6.9.
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 check the figure's angle for a drawn triangle.