At the aquarium where you work, a fish has gone missing in a 10-m-deep. 11-m-diameter cylindrical tank. You shine a flashlight in from the top edge of the tank, hoping to see if the missing fish is on the bottom. What's the smallest angle your flashlight beam can make with the horizontal if it's to illuminate the bottom?

To understand how to solve the given problem about illuminating the bottom of a tank, we need to be familiar with the concept of tangent relationship in trigonometry. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

This relationship can be expressed as: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). When applying this to practical problems like the one about the missing fish in the aquarium, we can define \(\theta\) as the angle of incidence we are trying to find. Here, the 'opposite' side would be the depth of the tank, and the 'adjacent' side is the radius of the tank's bottom. It's crucial to note that this method is applicable only in right-angled triangles, and familiarity with this concept is essential for solving problems in physics, engineering, and even daily life where angles and distances are involved.

To ensure students fully understand this concept, it's helpful to encourage them to draw the scenario, label the sides of the triangle, and actively use the relationship by computing examples with different values.

To find an angle in a right triangle when the lengths of two sides are known, one must use inverse trigonometric functions. These functions are the inverses of the sine, cosine, and tangent functions, and they allow one to calculate the measure of an angle given certain trigonometric ratios. In the case of the aquarium problem, we specifically use the inverse tangent function, denoted as \(\tan^{-1}\) or arctan.

The formula is applied as follows: \(\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\). Using this formula with the given values, one can determine the angle of incidence of the flashlight's beam. For practical computations, we can use scientific calculators or computational software that have these inverse functions built-in, allowing for the direct input of the values to receive the angle as the output.

To help students grasp this concept, introduce them to the different inverse trigonometric functions and provide several examples they can work through. Encourage the use of these functions not only in theoretical problems but also in real-world applications where angle measurements are necessary.

In physics and optics, the angle of incidence plays a significant role. This is the angle that a ray of light or another linear wave makes with a line perpendicular to the surface at the point of incidence. In the context of our aquarium problem, the angle of incidence refers to the angle between the flashlight beam and a line perpendicular to the water surface, which is essentially horizontal. Since the light travels in a straight line, a ray hitting the surface at this angle continues to the bottom of the tank at the same angle, assuming there are no refractive effects taking place at the water's surface.

The angle of incidence can be determined using the trigonometric relationships we've discussed. It's important to note that, in this question, finding the 'smallest' angle means we are looking for the critical angle at which the entire bottom of the tank starts to become illuminated, just as the light skims the edge.

Demonstrating this concept with a real beam of light and a model can significantly improve comprehension. This tangible example translates abstract mathematical concepts into a visual experience, reinforcing how angles influence the path of light - a concept fundamental to the fields of physics, astronomy, and engineering.