Graph – Sine and Cosine

INTRODUCTION TO PERIODIC FUNCTIONS

In mathematics, we often encounter functions that repeat periodically. Among the most important representatives of such functions are the sine and cosine functions. Both belong to the trigonometric functions, whose main characteristic is periodicity, as their values repeat at regular intervals. This means there exists a constant T for which f(x + T) = f(x) for all values of the variable x.

SHAPE AND PROPERTIES OF THE SINE FUNCTION

The sine function is usually written as f(x) = sin(x). Its graph has a wave-like shape that starts at the origin (0, 0), rises to a maximum value of 1 at x = π/2, returns to the origin at x = π, then falls to a minimum value of -1 at x = 3π/2, and completes one full period at x = 2π. The period of the function is therefore 2π. The values of the function are bounded between -1 and 1, meaning its range is the interval [-1, 1].

SHAPE AND PROPERTIES OF THE COSINE FUNCTION

The cosine function is given as f(x) = cos(x). Its shape is similar to the sine function, but its graph starts at the maximum value of 1 at x=0, reaches 0 at x = π/2, falls to -1 at x = π, reaches 0 again at x = 3π/2, and completes its period at 1 when x = 2π. Here too, the period is 2π, and the values also lie in the interval [-1, 1]. This function is even, meaning cos(-x) = cos(x).

GENERAL FORM AND PARAMETERS

Both functions can be written in a more general form:
f(x) = A * sin(Bx + C) + D or f(x) = A * cos(Bx + C) + D.

The parameters have the following meanings:

• A determines the amplitude, i.e., the maximum deviation from the central axis (if A = 2, then the graph oscillates between -2 and 2, assuming D=0).
• B affects the periodicity: the new period becomes 2π / |B|.
• C influences the horizontal shift (phase shift). A positive C often indicates a shift to the left, but this depends on how Bx+C is factored (e.g., B(x + C/B)).
• D shifts the graph vertically up or down (vertical shift).

EXAMPLE OF CALCULATION AND GRAPH

Let's take the function f(x) = 2 * sin(x – π/2) + 1.
Here:

• A = 2 → amplitude is 2.
• B = 1 → period remains 2π.
• The term (x – π/2) indicates a phase shift. C (if thinking Bx+C) would be related to -π/2. Specifically, x – π/2 means the graph is shifted to the right by π/2.
• D = 1 → the entire graph is shifted up by 1 unit.

The graph of this function, therefore, ranges between 2*(-1)+1 = -1 and 2*(1)+1 = 3, and has the same basic shape as the sine function but is shifted and stretched vertically.

CONCLUSION

The discussed graphs are fundamental examples of periodic functions in mathematics. Their regular shape, symmetry, and clear structure allow for easy analysis and transformation, enabling a deeper understanding of their behavior in various contexts.