Trigonometry is used in optics, the study of the behavior of light. The phenomena of most interest are reflection and refraction. A light ray changes direction when it is reflected from a mirror or smooth, shiny surface. If a ray of light passes from one transparent medium into another, the ray may be bent; this is refraction.
Any smooth, shiny surface reflects some of the light that strikes it. If the surface is perfectly flat, perfectly shiny, and reflects all of the light that strikes it, then any ray that encounters the surface is reflected away at the same angle at which it hits. You have heard the expression, “The angle of incidence equals the angle of reflection. ” This principle, known as the law of reflection, is illustrated in Fig. 10-1.
Flat and Non-Flat Surfaces
In optics, the angle of incidence and the angle of reflection are conventionally measured relative to a line normal (perpendicular) to the surface at the point where reflection takes place. In Fig. 10-1, these angles are denoted θ, and can range from 0°, where the light ray strikes at a right angle with respect to the surface, to almost 90°, a grazing angle relative to the surface. Sometimes the angle of incidence and the angle of reflection are expressed relative to the surface itself, rather than relative to a normal line.
If the reflective surface is not perfectly flat, then the law of reflection still applies for each ray of light striking the surface at a specific point. In such a case, the reflection is considered relative to a line normal to a flat plane passing through the point, tangent to the surface at that point. When many parallel rays of light strike a curved or irregular reflective surface at many different points, each ray obeys the law of reflection, but the reflected rays do not all emerge parallel. In some cases they converge; in other cases they diverge. In still other cases the rays are haphazardly scattered.
Reflection Practice Problems
Imagine a room that measures exactly 5.000 meters square, with one mirrored wall. Suppose you stand near one wall (call it “wall W ” as shown in Fig. 10-2A), and hold a flashlight so its bulb is 1.000 meter away from wall W and 3.000 meters away from the mirrored wall. Suppose you aim the flashlight horizontally at the mirrored wall so the center of its beam strikes the mirror at an angle of 70.00° relative to the mirror surface. The beam reflects off the mirror and hits the wall opposite the mirror. The center of the beam strikes the wall opposite the mirror at a certain distance d from wall W. Find d to the nearest centimeter.
The path of the light beam is in a plane parallel to the floor and the ceiling, because the flashlight is aimed horizontally. Therefore, we can diagram the situation as shown in Fig. 10-2B. The center of the beam strikes the mirror at an angle of 20.00° relative to the normal. According to the law of reflection, it also reflects from the mirror at an angle of 20.00° relative to the normal. The path of the light beam thus forms the hypotenuses of two right triangles, one whose base measures e meters and whose height is 3.000 meters, and the other whose base measures f meters and whose height is 5.000 meters. If we can determine the values of e and f , then we can easily get the distance d in meters.
- sing the right-triangle model for the tangent function, we can calculate e as follows:
We calculate f in a similar way:
Knowing both e and f , we calculate d, in meters, as follows:
To get d to the nearest centimeter, multiply by 100 and round off:
d = 3.91176 × 100 = 391 centimeters
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