Problem 237 - Precalculus

Question

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

Step-by-Step Solution

Answer

The largest possible value of \( f(x) \) is \( \frac{3}{5} \).

Step 1: Understanding the Cosine Function

The cosine function, \( \cos(x) \), oscillates between -1 and 1 for all \(x\). Hence, for any argument \(u\), \( \cos(u) \) takes its maximum value of 1.

Step 2: Applying the Property to \(f(x)\)

The function given is \( f(x) = \frac{3}{5} \cos(6x) \). To find the maximum value of \(f(x)\), we need to find the maximum value of \( \frac{3}{5} \cos(6x) \). Since the maximum value of \( \cos(6x) \) is 1, the maximum value of \( f(x) \) will be \( \frac{3}{5} \times 1 \).

Step 3: Calculating the Maximum Value

Multiply the coefficient outside the cosine function by the maximum value of the cosine. Thus, \( \frac{3}{5} \times 1 = \frac{3}{5} \).

Step 4: Conclusion

The largest possible value of \( f(x) \) is \( \frac{3}{5} \). This is obtained when \( \cos(6x) \) equals 1.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(x) \), is one of the fundamental trigonometric functions. It relates to the angles and sides of a right-angled triangle. The cosine function gives the ratio of the adjacent side over the hypotenuse. It is defined for all real numbers and often used in periodic functions such as waves.

• Cosine is periodic with a period of \( 2\pi \).
• Its range is between \(-1\) and \(1\).
• At \( \cos(0) \), its value is \(1\), which is its maximum value.

Being an even function means \( \cos(-x) = \cos(x) \), which is symmetrical to the y-axis. This property is useful when solving trigonometric equations, as it allows a straightforward understanding of behavior and symmetries involved.

Amplitude

Amplitude is a key concept when dealing with trigonometric functions, especially in contexts of oscillation and waves. In the function \(f(x) = \frac{3}{5} \cos(6x)\), the amplitude defines the extent to which the function oscillates above or below its central axis (usually the x-axis).

• The amplitude is determined by the coefficient in front of the trigonometric function.
• For \(f(x) = \frac{3}{5} \cos(6x)\), the amplitude is \(\frac{3}{5}\).

This signifies that the function will reach as high as \(\frac{3}{5}\) and as low as \(-\frac{3}{5}\) from its midline.

Maximum Value

In trigonometric functions, identifying the maximum value involves determining when the cosine function reaches its peak. For the function \(f(x) = \frac{3}{5} \cos(6x)\), understanding its amplitude helps here. Since the cosine function ranges between \(-1\) and \(1\), its peak is at \(1\).

• Multiply the maximum value of \(\cos(6x)\) by the amplitude.
• In this instance, \(\frac{3}{5} \times 1 = \frac{3}{5}\).

Thus, the maximum value of \(f(x)\) is \(\frac{3}{5}\), occurring when \(\cos(6x) = 1\).

Periodic Functions

Trigonometric functions are a classic example of periodic functions. These functions repeat their values in regular intervals over their domain.
Characteristics:

• The period of a function describes how frequently the function's shape repeats. For \(\cos(x)\), the standard period is \(2\pi\).
• The function \(f(x) = \frac{3}{5} \cos(6x)\) has its argument multiplied by \(6\), which compresses the period to \(\frac{2\pi}{6} = \frac{\pi}{3}\).

These features are beneficial in numerous applications, such as modeling sound waves, tidal waves, and other cyclical patterns encountered in nature and engineering.