Question

Identify the amplitude and period of the following functions. $$p(t)=2.5 \sin \left(\frac{1}{2}(t-3)\right)$$

Short answer

Answer: The amplitude is 2.5 and the period is $$4\pi$$.

Step-by-step solution

Identify the coefficient of the sine function

In the given function, $$p(t) = 2.5 \sin \left(\frac{1}{2}(t-3)\right)$$, the coefficient of the sine function is 2.5.

Find the amplitude

Since the amplitude is the absolute value of the coefficient of the sine function, we find the amplitude to be $$|2.5| = 2.5$$.

Identify the coefficient of the variable inside the sine function

In the given function, $$p(t) = 2.5 \sin \left(\frac{1}{2}(t-3)\right)$$, the coefficient of the variable t inside the sine function is $$\frac{1}{2}$$.

Calculate the period

The period of a sinusoidal function is given by $$\frac{2\pi}{B}$$, where B is the coefficient of the variable inside the sine function. In our case, B = $$\frac{1}{2}$$. Therefore, the period is $$\frac{2\pi}{\frac{1}{2}} = 2\pi \cdot 2 = 4\pi$$. The amplitude and period of the function $$p(t) = 2.5 \sin \left(\frac{1}{2}(t-3)\right)$$ are as follows: Amplitude: 2.5 Period: $$4\pi$$

Key concepts

Amplitude

The concept of amplitude is crucial when studying trigonometric functions like the sine function. Amplitude refers to the height of the wave measured from the midline to the peak. It essentially tells us how "tall" the wave is.

• The amplitude is always a positive number or zero.
• In a function like \(a \sin(bx + c)\), the amplitude is represented by the absolute value of \(a\).

In the function \(p(t) = 2.5 \sin \left(\frac{1}{2}(t-3)\right)\), the sine coefficient is 2.5. Therefore, the amplitude is the absolute value of this coefficient, which is \(|2.5| = 2.5\).
Understanding amplitude helps in visualizing how intense or minimized the oscillations of the sine wave are. Larger amplitudes mean a more "extreme" curve in terms of high and low peaks.

Period

The period of a sine function is the length of one complete cycle of the wave. This cycle tells you how long it takes for the function to repeat itself.

• The standard period of sine and cosine functions is \(2\pi\), but this can be modified by the coefficient in front of the variable.
• In a function \(a \sin(bx + c)\), the period is calculated by \(\frac{2\pi}{b}\).

In the given function \(p(t) = 2.5 \sin \left(\frac{1}{2}(t-3)\right)\), the coefficient \(b\) before \(t\) is \(\frac{1}{2}\). So, the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).
Knowing the period helps you identify how often the wave pattern repeats. A longer period means the waves are stretched out horizontally, occurring less frequently.

Sine Function

The sine function is one of the basic trigonometric functions. It is known for its wave-like pattern and is instrumental in modeling periodic phenomena. The general form of a sine function is \(a \sin(bx + c) + d\).

• \(a\) affects the amplitude of the sine wave.
• \(b\) influences the period, determining how stretched or compressed the wave is along the axis.
• \(c\) shifts the graph horizontally, known as the phase shift.
• \(d\) moves the graph up or down, affecting the vertical shift.

In \(p(t) = 2.5 \sin \left(\frac{1}{2}(t-3)\right)\), the transformations can be broken down as follows:- The amplitude is 2.5, indicating how high the wave peaks.- The period is \(4\pi\), indicating how long one complete wave cycle takes.- The \(\left(t-3\right)\) inside the sine moves the wave to the right by 3 units, which is the phase shift.
This function serves as a perfect example of how trigonometric functions can model repetitive structures and events, from sound waves to seasonal temperatures.