At a glance. One lesson of about 60 minutes. It assumes sine, cosine and tangent from Year 9. Students draw triangles from words, compare elevation and depression, choose the ratio, decide whether to multiply or divide, and solve two-triangle problems.
Calculator: GDC optional. A calculator is used for the ratios, so the label is GDC optional. Any scientific calculator does the same work, and the closed section covers degree mode and the inverse keys.
What students will learn
Know
- The angle of depression from A to B equals the angle of elevation from B to A.
- Opposite, adjacent and hypotenuse are named from the marked angle.
- Multiply when the wanted side is the numerator of the ratio and divide when it is the denominator.
Understand
- Both angles are measured from a level line, so they are alternate angles between parallel lines.
- Which operation to use is decided by which side is the unknown, not by the ratio's name.
Do
- Draw and label the triangle for a worded problem.
- Find a side or an angle with the correct ratio.
- Combine two right triangles when two angles are given.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: choosing and using sine, cosine and tangent, and the elevation and depression angles (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe what stays the same when height or angle changes and what makes a ratio multiply or divide.
- Criterion C. Communicating: a labelled sketch with the angle, the known side and the wanted side before any calculation (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the lookout, the lighthouse and the kite.
The one thing to do with the figure
Make students predict, then reveal. Before touching the first figure, ask each student to sketch the two angles and say whether they match. Then set the sliders to values of their own choosing and check.
In the second figure, set the angle once, then step through the six pairs. Ask: which pairs share the same ratio? (1 and 4 share sin; 2 and 6 share cos; 3 and 5 share tan.) What differs within a pair?
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the lookout and the boat. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 10 min | Second figure; the three worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to which side is the unknown. |
| 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 9. |
The answers
| Warm-up | Sine; 37°; 53.5. |
|---|---|
| Worked examples | 38.4 m; 72.0 m; 13.1°. |
| Faded example | 32.0 m. |
| Something to think about | More than double: about 2.14h to 5.67h. |
| 1. Choosing multiply or divide | B. A multiplies where the unknown is on the bottom; C finds the line of sight; D treats 15 as the hypotenuse. |
| 2. Height from an angle of elevation | 23.9 m. 45 × tan 28°. |
| 3. Angle of a ramp | 7.6°. tan−1(1.2 ÷ 9). |
| 4. Distance from an angle of depression | 81 m. 36 ÷ tan 24° = 80.9. |
| 5. Line of sight | 139.5 m. 50 ÷ sin 21°. |
| 6. Height of a flagpole | 17.1 m. 70 tan 41° − 70 tan 32°. |
| 7. Two boats in line | 203 m. 298.56 − 95.34. |
| 8. Walking towards a tower | 30.3 m. x ≈ 43.3, then h = x tan 35°. |
| 9. Height of a kite | A. B uses cosine; C uses tangent; D divides. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Multiplied by tan, found the line of sight, or used the hypotenuse formula. | Mark the 15 m as opposite the angle at the car and write tan 32° = 15 ÷ d before any key press. |
| 84.6 or 21.1 in question 2 | Divided by tan, or used sine with no hypotenuse. | Ask: is the wanted side on the top or the bottom of opposite ÷ adjacent? |
| 7.7 or 82.4 in question 3 | Used sine with the horizontal distance, or inverted the fraction. | Write opposite ÷ adjacent first, with the numbers in it, and only then press the inverse key. |
| 16 or 89 in question 4 | Multiplied by tan, or found the line of sight. | Ask which side the 36 m is, and which side the distance along the water is. |
| 130.3 or 17.9 in question 5 | Found the distance on the water, or multiplied by sine. | Mark the hypotenuse in a different colour in the sketch. |
| 60.9 or 11.1 in question 6 | Gave the top's height, or subtracted the angles. | Ask: from the ground, how high is each of the two points? Then subtract the heights. |
| 172 or 299 in question 7 | Subtracted the angles, or gave only the far boat. | Draw two triangles with the same vertical side, in two colours. |
| 28 or 14.6 in question 8 | Used the 40 m as the whole distance. | Name the unknown distance x, and write h in terms of x twice. |
| B, C or D in question 9 | Used cosine, tangent, or divided. | Say aloud: the string is the hypotenuse, the height is opposite. |
Questions worth asking
- Why must the angle of elevation at the boat equal the angle of depression at the lookout? Listen for “the horizontal lines are parallel, so the angles are alternate”.
- If you know the angle and one side, how many ratios could you use? Listen for “one for each unknown, and the choice depends on which two sides are involved”.
- Should the answer be bigger or smaller than the side you know? How can you tell before calculating? Listen for “the hypotenuse is the longest side”.
- Why can two angles of elevation from the same point not just be subtracted? The tangent is not linear: tan 41° − tan 32° is not tan 9°.
Differentiation
A way in for anyone stuck
- Give a labelled triangle template with the three sides named and a box for the ratio; the student fills in known and wanted.
- Provide a rule card: “unknown on top: multiply. Unknown on the bottom: divide.”
- Restrict to the first four questions at first, using whole-number angle and side values.
Stretch
- Show that a lookout of height h that sees two boats in line, at depressions a and b, finds them h(1 ÷ tan a − 1 ÷ tan b) apart.
- A tower's top is seen at elevation a from a point a distance d from its foot, and at elevation b from a point 2d from its foot. Show that tan a = 2 tan b.
Students learning in English as an additional language
“Elevation” means going up and “depression” means going down; use arm gestures. “Opposite” and “adjacent” are named from the marked angle, not from the page. Say “sine of 36 degrees” aloud and point to the two sides it links.
| Word | Plain meaning | Example |
|---|---|---|
| angle of elevation | the angle measured up from a level line to a line of sight | looking up from a harbour to a hilltop |
| angle of depression | the angle measured down from a level line to a line of sight | a bird looking down at a fish |
| line of sight | the straight line from the eye to the thing being looked at | a laser pointer's beam |
| horizontal | level, parallel to the ground | the surface of still water |
| opposite | the side across the triangle from the marked angle | the side not touching the angle |
| inverse tangent | the key that finds an angle from a tangent ratio | tan−1 on the calculator |
Sentence starters for explaining aloud or in writing:
- The angle of depression equals the angle of elevation because …
- The known side is the … and the wanted side is the …, so I use …
- I multiply (or divide) because the unknown is on the top (or bottom) of the ratio.
Two things not to say
“SOH CAH TOA, just plug in.” Ask the student to name the sides before choosing the ratio.
“Depression is the angle from the cliff face.” It is from the horizontal.
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.
- ElevationA 12 m pole casts a shadow 9 m long. Find the angle of elevation of the sun, to the nearest degree.
53°. - Earlier work (Year 9)A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side.
12 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA kite flies at the end of 60 m of string at an angle of 55° with the ground. How high is it above the ground?
49.1 m. - Working backFrom the top of a 40 m tower, the angle of depression of a car is 18°. How far is the car from the foot of the tower, to the nearest metre?
123 m.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why a 35° angle of depression and a 35° angle of elevation of the same two points are the same measurement.
They are alternate angles between the two horizontal lines. - OpenTwo boats are 100 m apart in line with a cliff, with angles of depression 20° and 30°. Find the cliff's height.
About 98.5 m.
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. “Show me the steps” gives the same states in words.
No device? A metre ruler and a protractor on a window sill give elevation angles outdoors. The With a GDC section is optional.