How to Find the Rate of Change of an Angle in Trigonometry

To find the rate of change of the angle formed by the vertical and the line of sight from the spotlight to the object, we need to use trigonometry. Let's call the angle formed by the vertical and the line of sight θ.

Since the distance between the spotlight and the object is decreasing, we can use similar triangles to relate the change in distance with the change in angle. First, let's set up the relationship between the distance and the angle:

tan(θ) = (12 m)/(13 m)

Now, let's differentiate both sides of the equation with respect to time (t):

sec^2(θ) * dθ/dt = 0 - (12/13^2) * (10 m/s)

After solving the equation, we get:

dθ/dt = -120/169 rad/s

So, the rate of change of the angle formed by the vertical and the line of sight from the spotlight to the object is approximately -0.709 rad/s.