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To appreciate the Beauty of Trigonometry, it is essential to understand the key feature of the trigonometric functions Cos[t] and Sin[t]:
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The trigonometric functions Cos[t] and Sin[t] provide a parametrization of the unit circle. But that is just the beginning—most importantly, a deep understanding of the sphere and torus is impossible without Cos[t] and Sin[t].
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The animation below is my attempt to illustrate how the parametrization of the unit circle works.
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The main components of the animation are as follows:
- The black circle is the unit circle.
- The values of t $\in [0,2\pi]$ are represented by orange lengths. In each scene there are three orange lengths: the horizontal length, the vertical length and the arc length on the unit circle.
- The values of Cos[t] are green horizontal line segments.
- The values of Sin[t] are blue vertical line segments.
- The principle above is the general principle of parametric equations of curves in the plane. Given two functions (one green and one blue) of the same independent variable t, if these functions define the coordinates of a moving point, then as the independent variable t varies, the point traces a curve in the plane.
Hover over the image to start the animation.
- The information sheet
- We will use
- To get started with Mathematica see my Mathematica page. Please watch the videos that are on my Mathematica. Watching the movies is essential for being able to organize your homework notebooks well! I will not discuss the basics of notebook structuring in class.
- In the study of the computational software system Wolfram Mathematica, we will focus on symbolic and numerical calculations, visualization, and programming in Mathematica. But Mathematica is much more than that. To read more about Mathematica see An Elementary Introduction to the Wolfram Language.
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As we begin the Mathematica portion of this class, a natural question arises: Why learn Mathematica? This is not an easy question to answer, but below are some key aspects that make Mathematica worth your attention.
- Mathematica is a powerful computational tool that can handle a wide range of mathematical problems, from basic algebra to advanced calculus and beyond.
- Unlike many other mathematical software systems, Mathematica excels in symbolic computation, allowing you to manipulate mathematical expressions and equations symbolically, closely resembling abstract mathematical thinking and structures.
- Mathematica offers powerful visualization tools, allowing you to create compelling graphics that enhance the understanding of mathematical concepts and effectively illustrate abstract mathematical results, fostering a varied and engaging mathematical learning experience.
- Mathematica is a high-level programming language that offers an extensive array of predefined functions, designed to simplify and streamline tasks that are commonly encountered in mathematical and technical computing. My impression is that this makes it more accessible and user-friendly compared to other mathematical software, enabling users to achieve complex objectives with greater ease.
- A lot of help is available on-line, for example Mathematica Stack Exchange, Wolfram Community and ChatGPT is quite good at offering help with Mathematica questions.
- Mathematica makes solving mathematical problems more efficient. How? For problems involving symbolic manipulation, we can perform symbolic manipulations in Mathematica exactly. For problems in which symbolic manipulations are inefficient, numerical approximations can be performed. Additionally, powerful graphical visualizations help guide the problem-solving process, offering deeper insights.
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I have enjoyed the interaction between my mind and computers since I first started using them in the early 1990s. Recently, I came across a powerful metaphor for this relationship, expressed by Steve Jobs. In an old interview, he illustrated the usefulness of computers with a compelling analogy: "For me, a computer has always been a bicycle for the mind." That perfectly describes Mathematica and other modern tools like ChatGPT. Isn't that wonderful?
See the Bicycle for the Mind video. Or, a longer
version.
- I am convinced that your mathematical learning experience will be enriched if you engage in mathematical explorations using technology.
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I recently encountered a fascinating geometric fact online: Imagine two squares where a vertex of the larger square is positioned at the center of the smaller square. Regardless of how these two squares are positioned, the area of their intersection remains constant. My description might not fully capture the beauty of this fact — indeed, an animation is worth a thousand words. How am I to make an animation? I must use technology. Here is an animation created in Mathematica.
Hover over the image to start the animation.
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I should have left it as an exercise for you to figure out why what I claim in the preceding item is true. But, the solution supported by a Mathematica animation might indicate why using technology can be powerful tool in communicating Mathematics.
Hover over the image to start the animation.
