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.
