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