Problem 21 - Precalculus

Question

Find the maximum value of \(|r|\) and any zeros of \(r\). $$r=4 \cos 3 \theta$$

Step-by-Step Solution

Answer

The maximum value of \(|r|\) is 4, and the zeros of \(r\) occur when \(\theta = (2n + 1)\frac{\pi}{6}\) for \(n\) being any integer.

Step 1: Determining Maximum Value of |r|

The maximum value of the absolute value \(|r|\) is equivalent to the highest point of the function \(4 \cos 3 \theta\). The cosine has an amplitude, or range of peak values, from -1 to 1, so \(|4 \cos 3 \theta|\) has the range from -4 to 4. The maximum value of \(|r|\) would, therefore, be 4.

Step 2: Locating Zeros of r

To find the zeros of \(r\), we need to examine when \(4 \cos 3 \theta = 0\). This happens when the argument of the cosine function equals \((2n + 1)\frac{\pi}{2}\) for \(n\) being any integer. Therefore, \(3 \theta = (2n + 1)\frac{\pi}{2}\). Solving for \(\theta\), we get \(\theta = (2n + 1)\frac{\pi}{6}\), where \(n\) is an integer.

Step 3: Summarizing Results

Summarizing these results, we can say that the maximum value of \(|r|\) is 4, and the zeros of \(r\) occur when \(\theta = (2n + 1)\frac{\pi}{6}\) for all integer \(n\).

Key Concepts

Amplitude of a Trigonometric Function

The amplitude of a trigonometric function, such as the sine or cosine, is a critical concept that describes the function's peak value. It directly affects how 'tall' or 'short' the wave of the function appears. When we look at the general form of a cosine function, given by
\( A \cos (B\theta + C) + D \),
the coefficient \( A \) represents the amplitude. For positive values of \( A \), it depicts the maximum height above the horizontal axis that the wave reaches. Conversely, for negative values, it indicates how deep the wave dips below the axis.
In our exercise, the function is \( r = 4 \cos 3 \theta \) and therefore has an amplitude of 4. This indicates that the function's wave oscillates from a maximum of 4 units above to 4 units below the horizontal axis—ultimately giving us a range from \( -4 \) to \( 4 \), although the absolute value of \( r \) restricts us to non-negative values, making the effective amplitude 4.

Finding Zeros of Trigonometric Functions

Identifying the zeros of a trigonometric function is essentially finding the angles at which the function has a value of zero. These points are where the function intersects the horizontal axis on a graph. For the cosine function, which is represented by a repeating wave, zeros occur at regular intervals depending on the function's period and phase shift.

To locate the zeros, we start by setting the trigonometric expression to zero and solving for the variable. In the exercise \( r = 4 \cos 3 \theta \), we set \( 4 \cos 3 \theta = 0 \) and solve for \( \theta \). For the cosine function, zeros occur at \( \frac{(2n+1)\pi}{2} \) for integers \( n \), which adjusts based on any coefficient placed before \( \theta \). Hence, after dividing by 3, we find that the zeros of \( r \) are at \( \theta = (2n + 1)\frac{\pi}{6} \) for integer values of \( n \). Understanding the periodic nature of trigonometric functions helps us predict these zeros at every cycle.

Determining Maximum Value of Trigonometric Expressions

The maximum (or minimum) value of a trigonometric expression highlights the extreme points of the wave on a graph, which are of key interest in various applications. The maximum value is the highest point on the wave above the horizontal axis; for the expression \( |r| \) in our exercise, it is where the cosine function reaches its peak.

Given that the basic cosine function oscillates between -1 and 1, when a coefficient is involved, such as 4 in our example \( r = 4 \cos 3 \theta \), the maximum value adapts accordingly. The absolute value of \( r \), indicated by \( |r| \), is at its maximum when \( \cos 3 \theta \) is 1 (as cosine cannot exceed 1). Thus, multiplying by the coefficient 4, the maximum value of \( |r| \) in our exercise becomes 4. This value represents the peak amplitude of the wave and can be crucial for determining the behavior of trigonometric functions in practical scenarios, such as engineering, physics, and other scientific fields.