Trigonometry · Ages 14–15
Angles of elevation and depression
Both from the horizontal; they are alternate angles.
How to do it
Method
Both from the horizontal; they are alternate angles
50 × tan 32° ≈ 31.2 m
Another way
The shared side first
The sight line first, then the height
In full
Looking up at something from the horizontal, the angle between the line of sight and the horizontal is the angle of elevation; looking down, the angle of depression. Both are measured from the horizontal, never from the upright. 50 m from a tower's foot, the top is at an elevation of 32°: the tower is 50 × tan 32° ≈ 31.2 m high. From the top, the same point is at a depression of 32° too: the horizontal is parallel to the ground, so the two are alternate angles, equal. With the eye above the ground, add the eye's height. Two points of view: write each right-angled triangle's equation, such as h = x tan β = (x + d) tan α, and solve for x and h. Another way: find a shared side (a line of sight, a distance) in one triangle, then use it in the other.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
From the ground 112 m from a tower's foot, the angle of elevation of its top is 52°. Correct to 1 decimal place, how tall is the tower, in m?
Show the answer and working
Answer: 143.4 m
tan 52° = height ÷ 112, so the height = 112 × tan 52° ≈ 143.4 m.
Example 2
Ming's eyes are 1.8 m above the ground, and he stands 147 m from a tree 45 m tall, looking at its top. To the nearest 0.1°, what is the angle of elevation?
Show the answer and working
Answer: 16.4°
The elevation is measured from his eye's horizontal: the top is 45 − 1.8 = 43.2 m above his eye, tan θ = 43.2147, θ ≈ 16.4°.
Example 3
From the ground 58 m from a tower 76 m tall, the angle of elevation of its top is more than 40°.
True or false?
Show the answer and working
Answer: True
The elevation = tan⁻¹(7658) ≈ 52.7°; compare it with 40°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.From point A, point B is at an angle of depression of 11°. What is the angle of elevation of A from B?
- A. 169°
- B. 11°
- C. 79°
- D. 22°
Choose the right answer
Answer:
2.From the top of a cliff 150 m high, a boat at sea is at an angle of depression of 43°. Correct to 1 decimal place, how far is the boat from the foot of the cliff, in m?
Answer:
3.From the ground 94 m from a tower's foot, the angle of elevation of its top is 58°. Correct to 1 decimal place, how tall is the tower, in m?
Answer:
4.From the ground 111 m from a tower 38 m tall, the angle of elevation of its top is more than 13°.
True or false?
Answer:
5.From the top of a lighthouse 94 m high, two boats in the same direction are at angles of depression of 45° and 14°. Correct to 1 decimal place: the nearer boat is m from the lighthouse's foot, and the boats are m apart.
Fill in each blank
Answer:
All the answers
- 1. B. 11°
- 2. 160.9 m
- 3. 150.4 m
- 4. True
- 5. 94, 283
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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