Transcendental functions aren't just about exponentials and logarithms - they also include trigonometric functions, which describe oscillations, periodic movements, and wave patterns.

This section explores how we can visualize these functions in Python with proper scaling, key points, and function behaviors.

Sine Function: Understanding Oscillations

Sine waves model natural oscillations, such as sound waves and circular motion. The sine function follows the general form:

How the Code Works

• Defines sine_function(x, a, b, c, d) to control amplitude (a), frequency (b), phase shift (c), and vertical shift (d);
• Generates x values over two full periods to capture the wave shape;
• Marks maxima, minima, and intercepts to highlight key points;
• Includes arrows at both ends to indicate the function continues indefinitely.

Cosine Function: A Phase-Shifted Sine Wave

Cosine functions behave similarly to sine but are phase-shifted by $\frac{\pi}{2}$. They are commonly used in oscillations, physics, and even electrical engineering.

How the Code Works

• Uses cosine_function(x, a, b, c, d) with the same parameters as sine;
• Marks key points:
  • Maxima at $x = 0$;
  • Minima at $x = \pm \pi$;
  • Intercepts where the function crosses zero.
• Adds arrows for infinite continuity.

Tangent Function: Dealing with Asymptotes

Tangent waves are different from sine and cosine because they have asymptotes at $x = \pm \frac{\raisebox{1pt}{$\pi$}}{\raisebox{-1pt}{$2$}}, \pm\frac{\raisebox{1pt}{$3\pi$}}{\raisebox{-1pt}{$2$}}$. These occur where $\cos(x) = 0$, making the function undefined.

How the Code Works

• Defines tangent_function(x) = tan(x);
• Splits x into three segments to avoid vertical asymptotes;
• Plots asymptotes as dashed red lines where the function is undefined;
• Includes arrows at both ends to show continuity;
• Adjusts zoom level to display only two asymptotes, avoiding graph clutter.