Motivation of sin and cos functions

In summary: 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. I think it would be difficult to use this approach to show that the arguments of the factors add up.It might be difficult to use this approach, but it is possible. I think the author of the book suggested using a different approach. He showed that the angle between two points in a plane is equal to the angle between the corresponding polar vectors. But in order to do that you need to parametrize an arc...and how do you do that without sine and cosine?It is possible to parametrize an arc
  • #1
Mappe
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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?
 
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  • #2
How are you defining the argument of a complex number without knowledge of sine and cosine?
 
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  • #3
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. I think it would be difficult to use this approach to show that the arguments of the factors add up.
 
  • #4
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.
But in order to do that you need to parametrize an arc...and how do you do that without sine and cosine?
Edit: Ah, wait I guess if you used y in terms of x, it would work. Never mind.
 
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  • #5
I don't understand your question as asked. I.e. how could one know the Taylor expansions before knowing the functions? To me the sin and cos functions are merely inverses of the arclength function. I.e. consider the unit circle, and define a function of y to be the arclength of the circle measured from the x-axis to the point of the circle at height y. This is a natural function. The inverse of this function is the sin function. I.e. given the arclength of a portion of the circle reaching from the x-axis to some point above it, (but remaining in the first quadrant), the y coordinate is the sin of that angle measured in radians. The cosine is similar.Having read it again, your question does not seem to be how to motivate sina nd cos but how to prove the angle of a complex product is the sum of the angles of the factors. I.e. how to relate x and y coordinates of points in the plane to their polar coordinates. But that is essentially the meaning of sin and cos.
 
  • #6
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?
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.
 
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Related to Motivation of sin and cos functions

1. What is the purpose or motivation behind the use of sin and cos functions in science?

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.

2. How do sin and cos functions relate to the concept of waves?

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.

3. What is the difference between sin and cos functions?

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.

4. How can sin and cos functions be used to solve real-world problems?

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.

5. Are there any limitations to using sin and cos functions in scientific research?

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.

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