Question

Identify the amplitude and period of the following functions. $$g(\theta)=3 \cos (\theta / 3)$$

Short answer

Answer: The amplitude is 3, and the period is \(6\pi\).

Step-by-step solution

Identify the parts of the function

The given function is in the form: $$g(\theta) = A\cos(\frac{\theta}{B})$$ Here, A and B are the parts of the function we are looking for. In our case, we have: $$A = 3$$ and $$B = 3$$

Find the amplitude

The amplitude is determined by the absolute value of the factor outside the cosine function. In our function, the factor outside the cosine function is 3. The amplitude is: $$\text{Amplitude} = |A| = |3| = 3$$

Find the period

The period of a cosine or sine function is given by: $$\text{Period} = \frac{2\pi}{|\text{Frequency Factor}|}$$ We can see that the frequency factor for the given function is the reciprocal of B, which is 1/3. $$\text{Frequency Factor} = \frac{1}{B} = \frac{1}{3}$$ Now, we can plug in the frequency factor to find the period: $$\text{Period} = \frac{2\pi}{|\frac{1}{3}|} = \frac{2\pi}{\frac{1}{3}} = 6\pi$$

Final answer

The amplitude of the given function \(g(\theta)=3\cos(\theta/3)\) is 3, and the period is \(6\pi\).

Key concepts

Amplitude

In trigonometric functions, the term "amplitude" refers to the maximum extent or height of the wave from its central axis. It indicates how much a trigonometric graph oscillates above and below its middle line.
Understanding amplitude is important when analyzing waves or signals, especially in fields like physics and engineering.

For the cosine function, amplitudes are determined by the coefficient that is in front of the cosine term. In the function given by the formula \(g(\theta) = 3 \cos(\theta/3)\), the coefficient is \(3\). Thus, the amplitude is \(|3| = 3\).

Key points to remember about amplitude:

• Amplitudes are always positive since it represents absolute distance.
• The larger the amplitude, the taller the peaks and the deeper the troughs of the wave.
• In graphs, amplitude alters the vertical stretching of the wave.

Recognizing amplitude is crucial to understanding how a function behaves as it reflects the magnitude of oscillation. Think of it as the way to gauge the "volume" or intensity of the wave.

Period

The concept of the period in trigonometric functions refers to the interval length after which the function levels repeat themselves. In essence, it represents the distance along the horizontal axis before a wave pattern starts to replicate.
When it comes to the cosine function, the formula to determine the period is \(\frac{2\pi}{|\text{Frequency Factor}|}\).

In the exercise \(g(\theta) = 3 \cos(\theta/3)\), we see \(\theta / 3\) within the cosine function. The frequency factor for this expression is computed as the reciprocal of \(B\) from \(\theta / B\), which is \(\frac{1}{3}\).
Therefore, the period becomes \(\frac{2\pi}{\frac{1}{3}} = 6\pi\).

Some essential thoughts about periods:

• They determine how "stretched" or "compressed" the wave appears horizontally.
• Smaller periods result in more cycles over a given interval.
• Period affects the spacings between equivalent points on each cycle.

Grasping how to calculate and identify the period helps understand how often waves repeat themselves. It's like knowing the "pace" of the wave's rhythm.

Cosine Function

The cosine function is one of the primary trigonometric functions and is essential in the study of angles and periodic phenomena. It describes a wave-like pattern that can model everything from sound waves to light oscillations.
Typically, the cosine function is written as \(A \cos(B\theta + C) + D\).

The standard cosine function \(\cos(\theta)\) has:

• An amplitude of \(1\)
• A period of \(2\pi\)
• A midline of \(y = 0\)

In different contexts, the cosine function can be transformed by altering the values of \(A\), \(B\), \(C\), and \(D\). In the exercise \(g(\theta) = 3 \cos(\theta/3)\), \(A\) is \(3\), providing the amplitude, and \(B\) is \(3\), affecting the period to become \(6\pi\) as per our calculations.

The cosine curve is symmetric about the vertical axis and begins at its maximum value when \(\theta = 0\). It moves in a repeated, regular interval making them excellent for modeling cycles.
Understanding the cosine function is pivotal. It's not just about graphs and waves but a tool to decipher real-world phenomena that manifest in cyclic patterns.