Trigonometric Functions – Sine

The sine function is one of the most fundamental concepts in mathematics, connecting geometry, trigonometry, and real-world applications. In its simplest form, sine is defined in a right triangle as the ratio of the length of the side opposite an angle to the hypotenuse. This simple ratio has far-reaching implications.

Opposite to Hpyotenuse is 1:2 with 30 degrees angle.

To understand sine more deeply, we can use the unit circle. The unit circle is a circle with a radius of one, centered at the origin. In this context, sine represents the y-coordinate of a point where the terminal side of an angle intersects the circle. As the angle increases, the sine value oscillates smoothly between negative one and positive one.

A:C is obviously equal to 0.53:1. The code for the animation of the unit circle is below.

Key angles like 30 degrees, 45 degrees, and 60 degrees provide exact values for sine. For 30 degrees, sine equals one-half. For 45 degrees, sine is approximately 0.71, which is the square root of two divided by two. For 60 degrees, sine equals 0.87, which is the square root of three divided by two. These values arise from special triangles, such as the 30-60-90 and 45-45-90 triangles.

By combining geometric and unit circle visualizations, sine becomes more than a mathematical concept. It transforms into a tool for solving problems in physics, engineering, and beyond. Mastering sine is not just about numbers; it is about seeing how mathematics describes the world. The sine function itself looks like this:

This is what happens, if we follow the animation of the unit circle in the YouTube video. The code is below.

This is the code, generating the unit circle:

And this is the simple code, making the sine function itself:

GitHub code – https://github.com/Vitosh/Python_personal/tree/master/YouTube/025-Trigonometric-Functions-Sine

Enjoy 🙂

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