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Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
But in order to do that you need to parametrize an arc...and how do you do that without sine and cosine?Fredrik said:In principle, the argument could be defined using the definition of "radian", but you'd have to use an integral to define the length of a curve segment.
Yes. The book "Visual Complex Analysis" has a very nice development in the first 15 pages that only establishes Euler's formula after the basic properties of the exponential, it's Tayler series, and the Taylor series of it's real and imaginary parts have been established. The sin and cos are almost afterthoughts.Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
Sin and cos functions are used in science to model and analyze periodic phenomena. This includes everything from the movement of planets in our solar system to the oscillation of sound waves. These functions help us understand and predict the behavior of these phenomena by representing them as smooth and continuous mathematical functions.
Sin and cos functions are closely related to the concept of waves. In fact, they are often used to describe the shape of waves, also known as waveforms. By plotting the values of a sin or cos function over time, we can see how it resembles the rise and fall of a wave, with its peaks and troughs.
The main difference between sin and cos functions is their starting point. A sin function starts at zero and increases to its maximum value, while a cos function starts at its maximum value and decreases to zero. This is because they are based on the unit circle, with the sin function corresponding to the y-coordinate and the cos function corresponding to the x-coordinate.
Sin and cos functions have many practical applications in fields such as engineering, physics, and astronomy. They can be used to model and analyze a wide range of phenomena, from the motion of a pendulum to the behavior of electrical circuits. By understanding the properties of these functions, scientists can make predictions and solve real-world problems.
While sin and cos functions are highly versatile and widely used in science, they do have some limitations. For example, they are based on the assumption of perfect periodicity, which may not always hold true in real-world situations. Additionally, these functions may not accurately represent non-linear or chaotic systems. It is important for scientists to carefully consider the limitations of these functions when applying them to research and data analysis.