Trigonometry is a powerful tool for measuring the heights of tall objects like buildings or trees. By standing at a distance and measuring an angle, we can calculate the height without needing to climb.
When we look up at a tall object, we create a right triangle. The height of the object is the opposite side, the distance from the observer to the base is the adjacent side, and the angle we measure is the angle of elevation.
The tangent function relates the angle to the ratio of opposite over adjacent sides. Since tangent of theta equals h over d, we can rearrange this to find h equals d times tangent of theta. This is our key formula for finding height.
Let's solve an example. A person stands 30 meters from a building and measures an angle of elevation of 40 degrees. Using our formula h equals d times tangent theta, we substitute: h equals 30 times tangent of 40 degrees, which is approximately 30 times 0.839, giving us about 25.2 meters.
This trigonometric method has many practical applications. It's used in architecture and construction to measure building heights, in surveying to map terrain, in navigation to determine distances, and even in astronomy to calculate celestial distances. Trigonometry makes measuring inaccessible heights both simple and accurate.