A lighthouse keeper 90 m above sea level observes a canoe. The angle of depression to the canoe is 18°. What is the horizontal distance from the canoe to the base of the lighthouse to 2 decimal places?

Grade 10 · Core Mathematics · Lesson 10 of 11

Applying Trigonometric Ratios to Angles of Elevation and Depression

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners calculate unknown horizontal distances from angles of depression using tangent ratios.

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L1040 minutes · 3 slides

KICD learning outcome (e)

Apply trigonometric ratios to angles of elevation and depression.

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

Calculate unknown horizontal distances from angles of depression using tangent ratios.
Determine angles of elevation and heights of structures using appropriate trigonometric ratios.
Solve contextual trigonometric problems involving ladders, towers, and right-angled triangles.

Key vocabulary

Angle of depression: Angle measured downward from a horizontal line of sight to an object.; A cliff observer sees a boat at an angle of depression of 22°.
Alternate angles: Equal angles formed on opposite sides of a line intersecting two parallel lines.; The angle of depression equals the angle of elevation because alternate interior angles are equal.
Angle of elevation: Angle measured upward from a horizontal line of sight to a higher target object.; Standing 350 m from a building, the angle of elevation to the roof is 52°.
Hypotenuse: The longest side of a right-angled triangle opposite to the right angle.; A 25 m ladder leaning against a wall forms the hypotenuse of the triangle.
Tangent ratio: The trigonometric ratio comparing the opposite side length to the adjacent side length.; A 100 m tower seen from 80 m away: tan θ = 100/80 = 1.25.

Lesson preview

Slide 1 · Calculating Horizontal Distance using Angle of DepressionModel · 12 min

Figure 1: A diagram of a 150 m watchtower and a vehicle, showing how the 22-degree angle of depression equals the alternate angle of elevation.

When looking down from a high point like a cliff or observation tower, the angle of depression is formed between the horizontal line of sight and the straight line of vision down to an object. By the alternate interior angles property of parallel horizontal lines, this angle equals the angle of elevation measured from the object up to the observer. We can set up a tangent ratio in the resulting right-angled triangle to solve for the horizontal distance across the ground or sea.

The angle of depression is measured downward from a horizontal line at the top observer point.
Horizontal parallel lines make the angle of depression equal to the angle of elevation from the object.
Setting $\tan θ = \frac{opposite}{adjacent}$ allows us to isolate the horizontal distance in the denominator.
Always maintain full calculator accuracy during evaluation before rounding to the requested decimal places.

Worked example. Let

x

be the horizontal distance in metres from the watchtower base to the vehicle. By alternate interior angles, the angle of elevation from the vehicle to the tower top is

22^{\circ}

. Set up the tangent ratio:

\tan 22^{\circ} = \frac{150}{x}

. Rearrange to calculate

x

x = \frac{150}{\tan 22^{\circ}} = \frac{150}{0.404026} \approx 371.26 m

. Answer: $371.26 m$ .

Check question: A lighthouse keeper $90 m$ above sea level observes a canoe. The angle of depression to the canoe is $18^{\circ}$ . What is the horizontal distance from the canoe to the base of the lighthouse to 2 decimal places?

$276.99 m$
$29.24 m$
$291.25 m$
$94.63 m$

Show answer

$276.99 m$ . The horizontal distance $d$ is the adjacent side and the height is the opposite side. $\tan 18^{\circ} = \frac{90}{d}$ , so $d = \frac{90}{\tan 18^{\circ}} = \frac{90}{0.324920} \approx 276.99 m$ .

Slide 2 · Determining Heights and Inclination Angles in Geometric SituationsPractice · 14 min

Figure 2: A right-angled triangle representing a 25 m ladder leaning against a vertical wall, with its base positioned 10 m from the wall.

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This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

All 3 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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Applying Trigonometric Ratios to Angles of Elevation and Depression

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