Sine Functions and Waves \[ y = \sin x \]

Sine Functions and Waves

Theory

The sine function is one of the most important functions in trigonometry and is widely used to model periodic phenomena. A periodic function is a function that repeats its values at regular intervals. The sine function creates a smooth and continuous wave pattern, making it ideal for representing cycles and oscillations.

Concept Explanation

The basic sine function is written as:

\[ y = \sin x \]

This graph oscillates between the values 1 and −1. One complete wave cycle starts at zero, rises to a maximum, falls through zero to a minimum, and then returns to zero again.

Key points on the sine graph are:

• At \(0^\circ\), \(y = 0\)
• At \(90^\circ\), \(y = 1\) (maximum)
• At \(180^\circ\), \(y = 0\)
• At \(270^\circ\), \(y = -1\) (minimum)
• At \(360^\circ\), \(y = 0\), completing one full cycle

Steps or Formula

The general form of a sine wave is:

\[ y = a \sin(k(x \pm c)) \]

From this formula, we can determine the main characteristics of the wave:

• Amplitude: \(|a|\)
• Period: \(\dfrac{360^\circ}{k}\)
• Maximum value: \(|a|\)
• Minimum value: \(-|a|\)

The amplitude represents the height of the wave from the center line, while the period shows the horizontal length of one complete cycle.

Example Problem

Consider the function:

\[ y = \sin x \]

Here, the values are:

• \(a = 1\)
• \(k = 1\)

Using the formulas:

• Amplitude \(= |1| = 1\)
• Period \(= \dfrac{360^\circ}{1} = 360^\circ\)
• Maximum value \(= 1\)
• Minimum value \(= -1\)

On the graph, the amplitude can be seen as a vertical distance of 1 from the x-axis to the peak, and the period is the horizontal distance covering one full wave.

Final Answer

The function \(y = \sin x\) produces a regular periodic wave with an amplitude of 1 and a period of \(360^\circ\). Its values range from −1 to 1 and repeat consistently every full cycle. Sine waves are widely used in real-world applications, such as modeling sound waves in physics and audio engineering.