![]() We could use another geometric argument to derive trigonometric relations involving θ−90º, but it is easier to use a simple trick: since formulas (1.4)−(1.6) hold for any angle θ, just replace θ by θ−90º in each formula. Rotating an angle θ by 90º in the clockwise direction results in the angle θ−90º. Using Figure 1 to match up those corresponding sides shows that the point (− y, x) is on the terminal side of θ+90º when ( x, y) is on the terminal side of θ. The rotation of θ by 90º does not change the length r of its terminal side, so the hypotenuses of the similar right triangles are equal, and hence by similarity the remaining corresponding sides are also equal. ![]() Thus, the right triangle in QI is similar to the right triangle in QII, since the triangles have the same angles. This forces the other angle of the right triangle in QII to be θ. The ray of light that bounces off the surface is called the reflected ray, and. Notice that the complement of θ in the right triangle in QI is the same as the supplement of the angle θ+90º in QII, since the sum of θ, its complement, and 90º equals 180º. The incident ray comes in at an angle, which is called the angle of incidence. In Figure 1 we see an angle θ in QI which is rotated by 90º, resulting in the angle θ +90º in QII. For example, suppose we rotate an angle θ around the origin by 90º in the counterclockwise direction. To rotate an angle means to rotate its terminal side around the origin when the angle is in standard position. The two operations on which we will concentrate in this section are rotation and reflection. In this article, we will take a look at how certain geometric operations can help simplify the use of trigonometric functions of any angle, and how some basic relations between those functions can be made. The phenomenon occurs if the angle of incidence is greater than a certain limiting angle, called the critical angle. 1 The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal in the plane formed by.
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