91Ó°ÊÓ

The angle of incidence is the angle between the incoming light ray and the normal (an imaginary line perpendicular to the surface) at the point of contact on the mirror. In this exercise, the light ray hits the first mirror at an angle of incidence of \(30^\circ\). This angle is crucial because it determines how the light will behave as it meets the mirror surface. According to the law of reflection, the angle of incidence is equal to the angle of reflection. Therefore, after hitting the surface, the light ray will reflect off at the same \(30^\circ\) angle as it enters. This predictable behavior helps us calculate the path and the number of reflections the light ray will undergo.

The path of a light ray is the trajectory it follows after striking a reflective surface. In this scenario, the light ray bounces back and forth between two mirrors, creating a zig-zagging pattern. This pattern occurs because each interaction with the mirror surfaces follows the law of reflection. Imagining a series of tiny, identical triangles formed by each bounce helps us understand this path more clearly.

• The base of the triangle is equal to the distance between the mirrors (20 cm).
• The hypotenuse represents the distance the light ray actually travels as it moves from one mirror to the other and back again.
• The vertical component or shift is determined by the angle of incidence, affecting how far the ray moves up or down the mirror with each reflection.

Understanding the path of light is crucial for calculating how reflections bring the ray to the opposite end of the mirror.

The tangent function is a trigonometric function that relates the angles of a triangle to the ratios of its sides. In our example, the tangent of \(30^\circ\) plays a critical role in determining how far the light travels vertically with each reflection. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle:\[\tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}}\]Since we know the distance between the mirrors (the base of each triangle) and the angle, we can use the tangent function to calculate the height shift (vertical movement) each time the light reflects. Substituting values:\[x = 2 \times 20 \times \tan(30^\circ)\]This equation gives us the key value of the vertical movement by relying on tangent, showcasing why this function is useful for problems involving angles and distances.

Unit conversion is a method used to ensure all measurements are in the same units, which is essential for accurate calculations. In this exercise, the length of the mirrors is given in meters (1.6 m), while the distance between them is in centimeters (20 cm). To simplify calculations, we convert the length of the mirrors from meters to centimeters. This way, all calculations can be performed more straightforwardly and consistently. Here's how it is done:

• Length in meters: 1.6 m
• Conversion to centimeters: \(1.6 \text{ m} = 160 \text{ cm}\)

Converting these measures allows us to directly compare and compute values, ensuring nothing confuses or mismatches during the calculation process. This step is often one of the first done in a mathematics problem to set up a coherent and unified problem-solving environment.