Question

Determine the amplitude and period of each function without graphing. $$ y=\frac{4}{3} \sin \left(\frac{2}{3} x\right) $$

Short answer

Amplitude: \frac{4}{3}, Period: 3\pi

Step-by-step solution

- Identify the Amplitude

The amplitude of a sine function \(y = A \sin(Bx)\) is given by the absolute value of the coefficient A in front of the sine function. In the function \(y = \frac{4}{3} \sin \left(\frac{2}{3} x\right)\), the coefficient A is \(\frac{4}{3}\). Therefore, the amplitude is \|\frac{4}{3}\| = \frac{4}{3}\.

- Identify the Period

The period of a sine function \(y = A \sin(Bx)\) is calculated by the formula \frac{2\pi}{B}\, where B is the coefficient of x inside the sine function. In this case, the coefficient B is \frac{2}{3}\. Therefore, the period is \frac{2\pi}{\frac{2}{3}} = \frac{2\pi \cdot 3}{2} = 3\pi\.

Key concepts

Amplitude Calculation

The amplitude of a sine function represents the maximum distance from the midline (or equilibrium position) to the peak of the wave. It's basically how 'tall' the wave is.

To calculate the amplitude, you look at the coefficient in front of the sine function. For a general sine function of the form \(y = A \sin(Bx)\), the amplitude is given by the absolute value of \(A\).

In our exercise, the function is \(y = \frac{4}{3} \sin\left(\frac{2}{3} x\right)\). Here, the coefficient \(A\) is \(\frac{4}{3}\).

So, the amplitude is \|\frac{4}{3}\|, which simplifies to \(\frac{4}{3}\).

• Identify the coefficient in front of \(\sin\).
• Take the absolute value of this coefficient to find the amplitude.

Period Calculation

The period of a sine function determines how long it takes for the function to complete one full cycle of its wave.

To calculate the period, we use the formula \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function.

For the given function \(y = \frac{4}{3} \sin\left(\frac{2}{3} x\right)\), the coefficient \(B\) is \(\frac{2}{3}\).

Using the formula, the period is \(\frac{2\pi}{\frac{2}{3}}\). When you multiply, it becomes \(\frac{2\pi \cdot 3}{2}\), which simplifies to \(3\pi\).

• Identify the coefficient inside the \(\sin\) function.
• Plug that coefficient into the period formula \(\frac{2\pi}{B}\).
• Simplify the equation to find the period.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are widely used in fields such as geometry, physics, engineering, and computer science.

In this context, the sine function is one of the primary trigonometric functions. It is periodic, meaning it repeats its values in regular intervals. Sine functions are associated with wave-like patterns.

Here are some essential points to remember:

• **Sine Function**: \( \sin(x)\).
• **Amplitude**: Measures the height of the wave. For \(y = A \sin(Bx)\), the amplitude is \|A\|.
• **Period**: The distance before the function repeats. Calculated as \( \frac{2\pi}{B} \) for the standard sine function.

These properties help us analyze and predict behaviors in many real-world phenomena, such as sound waves, light waves, and even tides.

Understanding these functions and their characteristics is crucial for solving many mathematical and scientific problems. So, it's important to practice and become comfortable with identifying and manipulating different trigonometric functions.