A lookout atop a vertical cliff 650 m high observes a ferry at an angle of depression of 18°. Calculate the horizontal distance of the ferry from the base of the cliff to 2 decimal places.

2000.49 m. The angle of depression is 18°, which means the alternate interior angle (the angle of elevation from the ferry to the top of the cliff) is also 18°. Using the tangent ratio: 18° = (opposite)/(adjacent) = (650)/(d). Rearranging to solve for the horizontal distance d, we get d = (650)/( 18°) ≈ 2000.49 m.

Grade 10 · Core Mathematics · Lesson 11 of 11

Reflecting on Real-Life Applications of Trigonometry

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners apply angles of elevation and depression to solve real-world contextual problems involving unknown heights and distances.

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L1140 minutes · 2 slides

KICD learning outcome (f)

Reflect on the use of trigonometry in real-life situations.

By the end of this lesson, learners should be able to

Apply angles of elevation and depression to solve real-world contextual problems involving unknown heights and distances.
Select and use appropriate trigonometric ratios to determine angles of inclination, structural angles, and distances.
Reflect on how trigonometry is used in land surveying, construction, navigation, and everyday infrastructure.

Key vocabulary

Angle of elevation: The angle measured upwards between the horizontal line of sight and an observed object.; The angle of elevation to the cliff top is used to compute height.
Angle of depression: The angle measured downwards between the horizontal line of sight and an observed object.; From the cliff top, the angle of depression to the ferry is $18^{\circ}$ .
Trigonometric Ratio: A ratio of the lengths of two sides of a right-angled triangle.; The tangent ratio relates the vertical height of a building to horizontal distance.

Lesson preview

Slide 1 · Modelling Real-Life Heights and Slopes using TrigonometryModel · 18 min

Figure 1: Two right-angled triangles show how angles of elevation and depression relate to horizontal and vertical distances.

Angles of elevation and depression allow us to measure inaccessible heights and distances by framing them as right-angled triangles. By identifying known sides or angles, appropriate trigonometric ratios like sine, cosine, or tangent can be selected to solve real-world problems. Whether calculating the slope of a road climbing a hill or determining how far a vessel is from a cliff, trigonometry converts physical observations into accurate geometric calculations. Use Figure 1 to see how to frame these hill and cliff scenarios as right-angled triangles.

The angle of elevation and depression can be applied to solve unknown distances and heights in real situations.
Identifying opposite, adjacent, and hypotenuse sides determines whether to use sine, cosine, or tangent ratios.
Calculating road slope angles on hills requires comparing vertical elevation change with total incline distance.
The angle of depression from a cliff top equals the angle of elevation from the observed object.

Worked example. Identify the given dimensions: vertical height opposite the angle is

45 m

and road length along hypotenuse is

500 m

. Set up the sine ratio formula:

\sin θ = \frac{vertical height}{incline length} = \frac{45}{500} = 0.09

. Find the inverse sine of

0.09

θ = \sin^{- 1} (0.09) \approx {5.16}^{\circ}

. Answer: The angle of the road slope is ${5.16}^{\circ}$ ..

Check question: A lookout atop a vertical cliff $650 m$ high observes a ferry at an angle of depression of $18^{\circ}$ . Calculate the horizontal distance of the ferry from the base of the cliff to 2 decimal places.

$211.20 m$
$683.47 m$
$2000.49 m$
$2103.44 m$

Show answer

$2000.49 m$ . The angle of depression is $18^{\circ}$ , which means the alternate interior angle (the angle of elevation from the ferry to the top of the cliff) is also $18^{\circ}$ . Using the tangent ratio: $\tan 18^{\circ} = \frac{opposite}{adjacent} = \frac{650}{d}$ . Rearranging to solve for the horizontal distance $d$ , we get $d = \frac{650}{\tan 18^{\circ}} \approx 2000.49 m$ .

Slide 2 · Reflecting on Practical Trigonometric Applications in Daily LifeCheck · 22 min

Figure 2: Diagrams displaying the angle of elevation for a ladder leaning against a wall and an observer looking up at a building.

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Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Reflecting on Real-Life Applications of Trigonometry

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Trigonometry 1