Problem 238 - Precalculus

Question

For the following exercises, let \(f(x)=\frac{3}{5} \cos (6 x)\) What is the smallest possible value for \(f(x) ?\)

Step-by-Step Solution

Answer

The smallest possible value for \( f(x) \) is \(-\frac{3}{5}\).

Step 1: Understand the Function

The given function is \( f(x) = \frac{3}{5} \cos (6x) \). This is a trigonometric function where \( \cos (6x) \) is multiplied by \( \frac{3}{5} \).

Step 2: Identify Range of Cosine

The cosine function, \( \cos(\theta) \), ranges from -1 to 1 for all \( \theta \). Thus, the range of \( \cos(6x) \) is also from -1 to 1.

Step 3: Apply the Range

Since cos(6x) varies between -1 and 1, when multiplied by \( \frac{3}{5} \), \( f(x) \) will range between \( \frac{3}{5}(-1) \) and \( \frac{3}{5}(1) \).

Step 4: Calculate the Minimum Value

The smallest possible value of \( f(x) \) is when \( \cos(6x) = -1 \). Substitute this into \( f(x) = \frac{3}{5} \cos(6x) \):\[ f(x) = \frac{3}{5}(-1) = -\frac{3}{5}.\]

Key Concepts

Range of Trigonometric Functions

In trigonometry, functions like sine and cosine have specific ranges that capture all the possible values they can yield. The cosine function, denoted as \( \cos(\theta) \), has a range from -1 to 1. This means, no matter the angle \( \theta \) you input, the resulting value will always lie within this interval.

The same concept applies when you modify the input angle or scale the function. In the exercise, the function is \( f(x) = \frac{3}{5} \cos(6x) \). Here, \( \cos(6x) \) still fluctuates between -1 and 1, even though the input is multiplied by 6. Thus, the range of \( f(x) \) doesn't expand or contract; it's just that \( \cos(6x) \) outputs values within its inherent limits, scaled by the factor \( \frac{3}{5} \).

Cosine Function

The cosine function is one of the fundamentals in trigonometry, often written as \( \cos(x) \). It's periodic, creating a wave-like graph that repeats every \( 2\pi \) radians.

In mathematics and physics, the cosine function helps in modeling phenomena such as waves and oscillations. In the given function \( f(x) = \frac{3}{5} \cos(6x) \), the \( 6x \) inside the cosine increases the frequency of the wave. This means the function completes more cycles in a given interval, leading to a more rapid oscillation of values between -1 and 1. The coefficient \( \frac{3}{5} \) scales the amplitude of this wave, ensuring the vertical stretch of each cycle is contained within a smaller range.

Understanding these elements helps you predict and analyze behaviors specific to trigonometric functions, like the one in this exercise.

Calculating Minimum Value

To find the minimum value in a trigonometric function, it's crucial to understand where within its range the function reaches its lowest point. For the cosine function, the minimum value occurs at \( \cos(\theta) = -1 \).

In the exercise, we have \( f(x) = \frac{3}{5} \cos(6x) \). When \( \cos(6x) = -1 \), substituting this into the function gives the minimum possible value of \( f(x) \). Calculate as follows:

• Substitute \(-1\) for \( \cos(6x)\): \[ f(x) = \frac{3}{5} \times (-1) \]
• Simplify to find \( f(x) = -\frac{3}{5} \), which is the minimum value.

Understanding this calculation helps in analyzing trigonometric functions and finding key points, such as minimum and maximum values, by leveraging their inherent properties.