my question is find the solution of this

Question

A rotating light is located 12 feet from a wall. The light completes one rotation every 5 seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is 15 degrees from perpendicular to the wall.

Answer 1

Let the light be located at (0, 0) and the point of the light be (12, y). Let the angle be θ. Then, y = 12 tan θ

We are trying to determine dy/dt

dy/dt = dy/dθ dθ/dt

As the light completes one rotation every 5 seconds, f = 1/5, and ω = 2π f = 2π/5

As y = 12 tan θ, dy/dθ = 12 sec²θ.

You could now solve using a calculator. dy/dt = dy/dθ dθ/dt = 12 sec² 15° * 2π/5 = ft/sec

If you needed a closed-form solution, you would need to know sec 15° = 1/cos 15°

As cos 30° = √3/2, from the half-angle formula, cos 15° = √((1+√3/2)/2) = 1/2 √(2+ √3)

Thus, sec 15° = 1/(1/2 √(2+ √3)) = 2/√(2+ √3), and sec² 15° = 4/(2+√3) = , multiplying by the conjugate (2-√3)/(2-√3), 8 - 4 √3

Then, 12 sec² 15° * 2π/5 = 12(8 - 4 √3)2π/5 = 192π/5 - 96√3π/5