Trigonometric Function Analysis \[ y = 3 \sin 2x \]

Trigonometric Function Analysis: y = 3 sin 2x

Theory

Trigonometric functions are commonly used to describe periodic motion and wave patterns. One of the most important trigonometric functions is the sine function, which produces a smooth, repeating wave. By changing certain parameters, we can adjust the height and speed of this wave.

Concept Explanation

In this lesson, we analyze the function:

\[ y = 3 \sin 2x \]

This function is a transformed version of the basic sine function \(y = \sin x\). The graph oscillates between positive three and negative three, creating a taller and faster wave compared to the standard sine curve.

Some important points on the graph include:

• At \(0^\circ\), the value is \(0\)
• At \(45^\circ\), the value reaches \(3\)
• At \(90^\circ\), the value returns to \(0\)
• At \(135^\circ\), the value reaches \(-3\)
• At \(180^\circ\), the function completes half a cycle

Steps or Formula

The general form of a sine function is:

\[ y = a \sin(kx) \]

From this formula, we can determine the key characteristics:

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

For the function \(y = 3 \sin 2x\), we identify:

• \(a = 3\)
• \(k = 2\)

Example Problem

Determine the amplitude and period of the function:

\[ y = 3 \sin 2x \]

Solution:

• Amplitude \(= |3| = 3\)
• Period \(= \dfrac{360^\circ}{2} = 180^\circ\)
• Maximum value \(= 3\)
• Minimum value \(= -3\)

This means the wave completes one full cycle every \(180^\circ\), which is twice as fast as the basic sine function.

Final Answer

The function \(y = 3 \sin 2x\) has an amplitude of 3 and a period of \(180^\circ\). Compared to \(y = \sin x\), it oscillates twice as fast and reaches higher maximum and minimum values. Such functions are useful for modeling real-world oscillating systems, such as mechanical vibrations and mass-spring motions.