Problem 9 - Algebra 2 and Trigonometry

Question

Find the amplitude of each function. \(y=0.6 \cos x\)

Step-by-Step Solution

Answer

The amplitude of the function is 0.6.

Step 1: Identify the Form of the Function

The function given is in the standard form of a cosine function, which is \( y = a \cos x \). Here, \( a \) is the coefficient of the cosine function.

Step 2: Determine the Amplitude

The amplitude of a cosine function \( y = a \cos x \) is the absolute value of the coefficient \( a \). In this case, \( a = 0.6 \).

Step 3: Calculate the Amplitude

Since the amplitude is the absolute value of the coefficient \( a \), we find: \[ \text{Amplitude} = |0.6| = 0.6 \].

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, alongside sine and tangent.
It is commonly represented as \( y = \cos x \), where \( x \) is the angle in radians or degrees.
This function is known for its periodic nature, repeating its wave-like pattern over a cycle of \( 2\pi \) radians or \( 360^{\circ} \).

• Its graph oscillates between -1 and 1.
• The highest point on the wave is 1, and the lowest is -1.

When a number multiplies the \( \cos x \), it affects the characteristics of the function, such as amplitude (which we'll discuss later) but does not change the fundamental periodicity.

Standard Form of Trigonometric Functions

Trigonometric functions like sine and cosine can be expressed in a standard form. For the cosine function, this form is \( y = a \cos x \).
In this equation:

• \( y \) describes the output or the value of the function.
• \( a \) is the coefficient that affects the function's amplitude.
• \( \cos x \) represents the cosine of the angle \( x \).

The coefficient \( a \) determines how much the wave contracts or expands vertically.
A positive \( a \) maintains the cosine wave's orientation, while a negative \( a \) inverts it across the \( x \)-axis.

Absolute Value

The concept of absolute value is key in determining the amplitude of trigonometric functions.
In mathematics, the absolute value of a number refers to its non-negative magnitude, regardless of its direction on the number line.

• For any number \( a \), its absolute value is written as \( |a| \).
• If \( a \) is positive or zero, then \( |a| = a \).
• If \( a \) is negative, then \( |a| = -a \).

When assessing the amplitude in a trigonometric function like \( y = a \cos x \), we find the amplitude by calculating \( |a| \).
This ensures that the amplitude is always a positive quantity, reflecting the maximum height of the wave's peak above the x-axis.