What's the easiest way to remember how to draw the graphs for the common trigonometric functions, like sin cos and tan?

The easiest way to remember how to draw the graphs of the common trigonometric functions like sine (sin), cosine (cos), and tangent (tan) is by understanding the key characteristics of each function and using mnemonic devices or patterns to help you memorize them.

1. Sine (sin) Function:
The graph of the sine function starts at the origin (0,0) and repeats its pattern infinitely in both the positive and negative y-direction. Key points to remember:
- Amplitude: The peak value of the graph, which is the distance between the maximum and minimum points. It is always 1.
- Period: The length of one complete cycle of the graph. The period of the sine function is 2π.
- Zeroes: The points where the graph intersects the x-axis. The sine function has zeroes at 0, π, 2π, etc.
- Symmetry: The sine function is an odd function, which means it is symmetric about the origin.

To remember the shape of the sine graph, you can use the mnemonic "Sea Turtle":
- "S" shape for the graph of sine.
- "E" for the graph starts and ends at the origin.
- "A" for the amplitude.
- "T" for the period of 2π.
- "U" for the up and down movement of the graph.
- "R" for the repeated pattern.

2. Cosine (cos) Function:
The graph of the cosine function also starts at the maximum point (1,0) and repeats its pattern. Key points to remember:
- Amplitude: The same as the sine function, it is always 1.
- Period: Again, the same as the sine function, it is 2π.
- Zeroes: The cosine function has its zeroes at π/2, π, 3π/2, etc.
- Symmetry: The cosine function is an even function, which means it is symmetric about the y-axis.

To remember the shape of the cosine graph, you can use the mnemonic "Captain Hook":
- "C" shape for the graph of cosine.
- "A" for the amplitude.
- "P" for the period of 2π.
- "T" for the starting point at (1,0).
- "A" for the ascending and descending movement of the graph.
- "I" for the inverted symmetry.

3. Tangent (tan) Function:
The graph of the tangent function is different from sine and cosine. Some key points to remember:
- The tangent function has vertical asymptotes (lines that the graph approaches but never touches). These vertical asymptotes occur at odd multiples of π/2: π/2, 3π/2, 5π/2, etc.
- The graph exhibits a repeating pattern with a period of π (one-fourth of the period of sine and cosine).
- The tangent graph also has horizontal asymptotes at y = 1 and y = -1.

To remember the shape of the tangent graph, you can use the mnemonic "The Cobra":
- "T" shape for the graph of tangent.
- "H" for the horizontal asymptotes at y = 1 and y = -1.
- "E" for the even spacing between vertical asymptotes (odd multiples of π/2).
- "C" for the repeating pattern with a period of π.
- "O" for the open arms of the graph that approach the asymptotes.
- "B" for the bulging or curvature between the asymptotes.

Remember, practice is key to ensure you become proficient at drawing these graphs.