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Measuring the height of a mountain can be challenging, especially when it is too tall to climb or observe directly. One effective method involves using basic trigonometry, specifically the sine function, combined with angle of elevation measurements taken from a certain distance.
The Concept of Angle of Elevation
The angle of elevation is the angle between the horizontal line from the observer’s eye and the line of sight to the top of the mountain. By measuring this angle from a known distance, we can calculate the mountain’s height using trigonometry.
Using Sine to Find the Mountain’s Height
Suppose an observer stands a distance d from the base of the mountain and measures an angle of elevation θ. The height of the mountain, h, can be calculated using the sine function:
sin(θ) = h / (d + h)
However, since the observer’s line of sight extends from their eye level, and assuming the observer’s eye is at ground level, the formula simplifies to:
h = d * tan(θ)
But if we only know the angle of elevation and the distance, using sine directly involves considering the right triangle formed by the height, the distance, and the hypotenuse. The more accurate formula for the height is:
h = d * tan(θ)
Practical Example
Imagine you stand 500 meters from the base of a mountain and measure an angle of elevation of 30°. To find the height of the mountain:
- Distance, d = 500 meters
- Angle of elevation, θ = 30°
Calculate the height:
h = 500 * tan(30°)
Using a calculator, tan(30°) ≈ 0.577. Therefore:
h ≈ 500 * 0.577 = 288.5 meters
The mountain is approximately 288.5 meters tall.
Summary
By measuring the angle of elevation from a known distance, you can use trigonometry—specifically the tangent function—to estimate the height of a mountain. This simple yet powerful method is widely used in surveying and geography to measure inaccessible heights accurately.