Graphing Tutorial 1 (Graph - Sine and Cosine)

HOW TO DRAW THE GRAPHS OF SINE AND COSINE FUNCTIONS

When we talk about trigonometric functions, sine and cosine are among the most basic and important. Their graphs are key to understanding wave motion and periodicity in mathematics and physics. In this article, we will explain how to draw the graphs of these two fundamental functions.

BASIC CHARACTERISTICS

Before we start drawing, it's important to understand some basic characteristics of cosine and sine, as these will help us:

• PERIODICITY: Both sine and cosine are periodic functions. They both have a fundamental period of 2π. This means their values repeat every 2π units.
• AMPLITUDE: The amplitude of both functions is 1, which tells us that their values vary between -1 and 1.
• PHASE SHIFT: Sine and cosine are phase-shifted from each other by π/2 or 90 degrees. When sine is at its highest point (1), cosine is at zero, and vice versa. (They have the same value at π/4 or 45 degrees, and other angles as well).

GRAPH OF THE SINE FUNCTION (sin x)

First, let's look at drawing the graph of the sine function:

1. X-AXIS: Mark the x-axis, which represents the angle in radians. On it, mark key points such as 0, π/2, π, (3π)/2, 2π, and so on – continuing in increments of π/2.
2. Y-AXIS: Mark the y-axis, which in our case represents the values of the sine function. These values, due to their amplitude mentioned earlier, range between -1 and 1.
3. POINTS ON THE GRAPH: Mark key points on the graph, such as (0,0), (π/2,1), (π,0), ((3π)/2,−1), (2π,0).
4. DRAWING THE CURVE: Connect the points with a smooth curve that represents a wave.

GRAPH OF THE COSINE FUNCTION (cos x)

We draw the graph of the cosine function similarly to the graph of sine:

1. X-AXIS AND Y-AXIS: Just as with sine, mark the x-axis and y-axis. The y-axis will range from -1 to 1.
2. KEY POINTS: For cosine, the key points are (0,1), (π/2,0), (π,−1), ((3π)/2,0), (2π,1).
3. DRAWING THE CURVE: Connect the points, being aware that the cosine curve starts at 1 when x is 0, which is exactly opposite to sine (sine starts at 0).

CONCLUSION

The graphs of sine and cosine are fundamental for understanding many concepts in trigonometry and wave theory. Correctly drawing these graphs helps in visually understanding phase shifts, periodicity, and amplitude. By practicing drawing these functions, we can deepen our understanding of more complex trigonometric concepts.