Answer:
68.2°
Step-by-step explanation:
To solve this problem, we can use the tangent ratio, which is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.
(TOH)
In this case, we'll use the tangent ratio to find the angle of depression.
Given:
- Height of lights above the stage (opposite side): 25 feet
- Distance from the performer to the lights (adjacent side): 10 feet
We want to find the angle of depression, which is the angle formed by the horizontal line of sight and the line of sight from the performer to the lights.
Let [tex] \textsf{angle of depression be } \theta [/tex]
Using the tangent ratio:
[tex] \tan( \theta ) (T) = \dfrac{\textsf{opposite(O)}}{\textsf{adjacent(A)}} [/tex]
Substituting the given values:
[tex] \tan(\theta) = \dfrac{25}{10} [/tex]
We can use the inverse tangent function (arctan or tan⁻¹) to find this angle.
[tex] \theta = \tan^{-1}\left(\dfrac{25}{10}\right) [/tex]
Using a calculator:
[tex] \theta \approx 68.198590513648 [/tex]
[tex] \theta \approx 68.2^\circ [/tex]
Therefore, the angle of depression at which the lamps A and B should be set is approximately [tex]\boxed{68.2^\circ}[/tex].
