Answer:
Approximately \(53^{\circ}\).
Step-by-step explanation:
Refer to the diagram attached. Note that the following three segments form the three sides of a right triangle:
- the \(80\; {\rm m}\)-segment between point \({\rm T}\) and the base of the tower along the ground level,
- the \(60\; {\rm m}\)-segment representing the height of the tower, and
- the imaginary segment joining the top of the building and the point \({\rm T}\) (hypotenuse).
The goal is to find the measure of the angle \(\theta\) (\(0^{\circ} < \theta < 90^{\circ}\)) adjacent to the \(60\; {\rm m}\)-segment in this right triangle. Since the length of the two legs in this right triangle are known, the angle \(\theta\) can be represented in term of its tangent, which represents the ratio between the length of the "opposite" side and the "adjacent" side:
- Side opposite to this angle: \(80\; {\rm m}\).
- Side adjacent to this angle: \(60\; {\rm m}\).
\(\displaystyle \tan(\theta) = \frac{(\text{opposite})}{(\text{adjacent})} = \frac{80\; {\rm m}}{60\; {\rm m}}\).
The measure of this angle can be found using the arctangent function:
\(\begin{aligned}\theta &= \arctan(\tan(\theta)) \\ &= \arctan\left(\frac{80\; {\rm m}}{60\; {\rm m}}\right) \\ &\approx 53^{\circ}\end{aligned}\).
(Note that if the output is in radians, multiply the value by \((180^{\circ}) / (\pi)\) to obtain the measure in degrees.)
Hence, the angle in this question would be approximately \(53^{\circ}\).
