01 · Explore
Line of Sight and Angle of Elevation
To measure the height of a tall object like a minar or a tower, we first need to understand how our vision relates to geometric angles.
The line of sight is defined as the imaginary line drawn from the eye of an observer to the point in the object being viewed. When we look up at an object, such as the top of a building, our line of sight rises above the horizontal level.
The angle of elevation is the angle formed by the line of sight with the horizontal line when the object is above the horizontal level. In simpler terms, it is the angle through which you raise your head to look at the top of an object.
To solve problems involving heights, we represent the scenario using a right-angled triangle. The object (like a tower) is usually the vertical side, the distance from the observer is the base, and the line of sight is the hypotenuse.
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Question
If you are looking at a bird in the sky, is the angle formed with the horizontal an angle of elevation or depression?
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02 · Explore
Angle of Depression
When an observer is at a height and looks down at an object on the ground, a different angle is formed.
The angle of depression is the angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level. This occurs when we lower our head to look at an object, such as a girl on a balcony looking at a flower pot on the street.
In geometric problems, the horizontal line at the observer's height is parallel to the ground. Therefore, the angle of depression is equal to the angle of elevation of the observer's eye as seen from the object, due to the property of alternate interior angles between parallel lines.
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Question
Why is the angle of depression from a cliff to a boat equal to the angle of elevation from the boat to the cliff?
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03 · Explore
Calculating Heights of Vertical Objects
By knowing the distance from an object and the angle of elevation, we can calculate the object's height using trigonometric ratios.
In most height and distance problems, we use the tangent (tan) ratio because it relates the height (opposite side) to the distance from the base (adjacent side). The formula is tan θ = Opposite / Adjacent.
If the height of the observer is given, it must be subtracted from the total height of the object to find the 'opposite' side of the right triangle, or added back at the end if we are finding the total height from the ground.
Finding Tower Height
tan 60° = h / 15
A tower stands vertically. From a point 15 m away, the angle of elevation is 60°. Let height be h. tan 60° = √3. So, √3 = h / 15, which means h = 15√3 m.
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Question
Which trigonometric ratio is most commonly used when the hypotenuse is not involved?
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04 · Explore
Accounting for Observer Height
In some problems, the height of the observer is significant and must be included in the geometric model.
When the observer's height is provided (e.g., 1.5 m), the right triangle is formed starting from the observer's eye level, not the ground. The base of this triangle is at a height above the ground equal to the observer's height.
To find the total height of an object like a chimney, you calculate the vertical side of the triangle (AE) and then add the observer's height (BE).
Chimney Height Calculation
Total Height = AE + 1.5 = 28.5 + 1.5 = 30 m
An observer 1.5 m tall is 28.5 m from a chimney. Angle of elevation is 45°. In the triangle, tan 45° = AE / 28.5. Since tan 45° = 1, AE = 28.5 m. Adding the observer's height gives 30 m.
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Question
If a 1.2 m tall girl looks at a balloon, and the triangle calculation gives a height of 87 m, what is the balloon's actual height from the ground?
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