Problem 45 - Algebra 2

Question

Find the amplitude and period of \(y=-0.2 \cos \frac{\pi}{3} \theta\)

Step-by-Step Solution

Answer

The amplitude of the function \(y=-0.2 \cos \frac{\pi}{3} \theta\) is 0.2, and the period is 6.

Step 1: Identify the coefficient of the function

The equation given is \(y=-0.2 \cos \frac{\pi}{3} \theta\). The coefficient of the cosine function is -0.2 which determines the amplitude.

Step 2: Compute the amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient of the cosine term. In this case, we have \(A = |-0.2| = 0.2\). Therefore, the amplitude of the given function is 0.2.

Step 3: Identify the coefficient of θ

In the given equation, the coefficient of θ under the cosine function is \(\frac{\pi}{3}\). This value will affect the period of the function.

Step 4: Compute the period

The period of a cosine function is given by \(2\pi / B\), where \(B\) is the coefficient of θ. Here, \(B = \frac{\pi}{3}\). Therefore, the period will be \(2\pi / \frac{\pi}{3} = 6\).

Key Concepts

Amplitude

In trigonometry, the amplitude of a function measures how far its waves stretch vertically from its center line. It essentially describes the height of the peaks and the depth of the troughs from the midline of the graph.

For a basic cosine function, which is typically expressed as \(y = a \cos(b\theta + c) + d\), the amplitude is determined by the absolute value of the coefficient \(a\).

• If \(a\) is positive, the wave is standard, starting its cycle at the maximum peak.
• If \(a\) is negative, the wave is flipped, starting at the minimum.

For the function \(y = -0.2 \cos \frac{\pi}{3} \theta\), the coefficient \(-0.2\) indicates that the amplitude is \(0.2\). The negative sign signifies that the graph flips vertically, but this does not affect the amplitude which is always a positive measurement.

Period

The period of a trigonometric function defines the distance (along the horizontal axis) required for the function to complete one full cycle. The period is a critical feature because it tells us how frequently the waves of the function repeat.

For the typical cosine function, \(y = a \cos(b\theta + c) + d\), the period can be calculated using the formula \(\frac{2\pi}{b}\), where \(b\) affects the frequency of the wave.

• A larger \(b\) results in a shorter period, meaning more cycles fit on the same interval.
• A smaller \(b\) extends the period, so fewer cycles occur per unit distance.

In the equation \(y = -0.2 \cos \frac{\pi}{3} \theta\), the coefficient \(b\) is \(\frac{\pi}{3}\). Plugging this into the period formula \(\frac{2\pi}{b}\) gives us \(\frac{2\pi}{\frac{\pi}{3}} = 6\). Therefore, the period of the function is \(6\), indicating each wave cycle spans a length of 6 on the angle axis.

Cosine function

The cosine function is one of the primary trigonometric functions and is pivotal in various mathematical contexts, especially when describing periodic phenomena. This function maps an angle to the x-coordinate of a point on the unit circle.

Generally expressed as \(y = a \cos(b\theta + c) + d\), the function has several key properties.

• Amplitude: Dictated by \(a\), it represents vertical stretch or shrinkage. The standard cosine's amplitude is 1.
• Period: Decided by \(b\), it defines how long it takes for the wave to complete a cycle, usually \(2\pi\) for the basic cosine.
• Phase Shift: Controlled by \(c\), this term translates the wave horizontally, shifting it along the \(\theta\)-axis.
• Vertical Shift: Managed by \(d\), it moves the entire function up or down on the graph.

Understanding these components is crucial for graphing and interpreting cosine functions in mathematical, engineering, and physical applications. In the given context of \(y = -0.2 \cos \frac{\pi}{3} \theta\), the wave is flipped and stretched to half its usual height. The period indicates the wave repeats every 6 units along the axis.