Someone in our class asked “After reviewing the past lessons of trigonometric functions again, I was just wondering if there are any real world instances that would link to such specific functions, and why these applications relate to the ratios and angles in a right triangle. ” I think I should share my answer to this question to all of you. Below is my reply.
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That’s a great question!
In my lectures, I mentioned the connection of sine/cosine functions with oscillations and waves, but because of time constraints, I did not go into details (which require some physics background). Below I am going to use a concrete example to explain the connection.
We know the sine or cosine functions can be defined as ratios related to angles (which can be viewed as rotations, and whose reference angles can be put into right triangles). They are the coordinates of a point moving around a unit circle. If you tie a stone on one end of a rope, hold the other end and uniformly spin the rope such that the stone rotates about your hand uniformly on a circular trajectory with your hand as its center, then the stone is like the point on the unit circle, and the amount of its rotation (the angle) about your hand is proportional to time. If you do so in the dark, and then shed a parallel light along the plane of the circle toward a wall perpendicular to the light, the shadow of the stone on the wall will oscillate back and forth about the shadow of your hand. The distance between the shadows of the stone and your hand is simply a coordinate of the stone relative to your hand (the origin). The motion of the shadow of the stone then can be described by a sine or cosine function (note that a cosine can be written as a sine with a phase shift) of the time.
In the real world, this kind of back-and-forth oscillation as the shadow of the stone above is almost everywhere. For example, when you play piano, a point on a string hit by a key oscillates back and forth; when you play saxophone, a small air volume in the tube oscillates back and forth; when you go to a beach, the sea water goes back and forth; when you use a pendulum clock, the pendulum goes back and forth. Other examples include heart beating, breathing, blinking:-), and the list can go on and on. All these oscillatory phenomena can be described as sine/cosine functions of time. If we can use a simple sine or cosine function of time to describe the oscillation, we say the oscillation is a simple harmonic motion. In a real-world phenomenon, an oscillation, say a hand or head shake:-), may not follow a simple harmonic motion. However, thanks to Fourier analysis (mentioned in my lectures), we can decompose any complicated oscillation into the sum of many simple harmonic motions with different amplitudes and periods (or say frequencies). An analogy of Fourier analysis is like listening to an orchestra. What you hear is a mixed sound (a complicated oscillation), but that mixed sound is composed of the simpler forms of music from various instruments. Fourier analysis is a very important tool that is widely used in mathematical analysis, scientific computing, signal processing and image processing.
Hopefully the above explanations motivate you to appreciate the beauty and power of math:-)
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