Problem Setup
We need to find the angle of elevation ($\theta$) of a tower's top from a point on the ground.
- Tower Height (Opposite side) = $200\sqrt{3}$ m
- Distance from base (Adjacent side) = 200 m
Calculating Angle of Elevation
In a right-angled triangle formed by the tower, the ground, and the line of sight to the top, the tangent of the angle of elevation is the ratio of the tower's height to the distance from its base.
Using trigonometry:
$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $Substitute the given values:
$ \tan(\theta) = \frac{200\sqrt{3}}{200} $Simplify the expression:
$ \tan(\theta) = \sqrt{3} $To find the angle $\theta$, we look for the angle whose tangent is $\sqrt{3}$.
$ \theta = \arctan(\sqrt{3}) $The angle whose tangent is $\sqrt{3}$ is $60^\circ$.
$ \theta = 60^\circ $Therefore, the angle of elevation of the tower's top is $60^\circ$.
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