Problem 5 - Algebra 2 and Trigonometry

Question

Find the amplitude of each function. \(y=5 \cos x\)

Step-by-Step Solution

Answer

The amplitude is 5.

Step 1: Identify the Form of the Function

The function is given as \( y = 5 \cos x \). Functions of the form \( y = a \cos x \) have an amplitude, which is determined by the coefficient \( a \).

Step 2: Determine the Amplitude

In the function \( y = 5 \cos x \), the coefficient of \( \cos x \) is 5. The amplitude of a cosine function is the absolute value of this coefficient.

Step 3: Calculate the Amplitude

Take the absolute value of the coefficient to find the amplitude: \( |5| = 5 \). Thus, the amplitude of the function \( y = 5 \cos x \) is 5.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, often related to the study of right triangles and oscillatory motion. They include the sine, cosine, and tangent functions, among others. These functions help describe:

• Relationships between angles and sides of triangles.
• Periodic phenomena such as waves and circular motion.

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. Each function has unique characteristics: amplitude, period, and phase shift. Amplitude relates to the height of the wave peaks and troughs above and below an equilibrium position. It's essential in describing how tall or short the peaks of a wave are in graph form. You often encounter these functions in physics, engineering, and astronomy and they are crucial for analyzing various systems and patterns observed in real-life applications.

Cosine Function

The cosine function, represented as \( \cos(x) \), is one of the primary trigonometric functions. It's characterized by its wave-like shape, known as a cosine wave.
The general form of the cosine function is \( y = a \cos(bx + c) + d \), where:

• \( a \) is the amplitude that tells us the wave's height.
• \( b \) affects the period, determining how quickly the function completes its cycle.
• \( c \) shifts the wave horizontally, known as the phase shift.
• \( d \) shifts it vertically.

For our specific example, \( y = 5 \cos x \), the amplitude is directly given by the coefficient 5. This means the cosine wave will reach a maximum of 5 and a minimum of -5 along the y-axis, reflecting how far the wave rises and falls from its central position.

Absolute Value

The absolute value is a fundamental mathematical concept. It represents the non-negative value of a number without considering its sign. You can think of it as the distance from zero on the number line. The absolute value of a number \( x \) is denoted as \( |x| \).
To compute the absolute value:

• If \( x \) is positive, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \).
• \( |x| \) is always greater than or equal to zero.

In the context of finding the amplitude of functions such as \( y = 5 \cos x \), the absolute value helps ensure that we only consider the magnitude of the coefficient. Thus, for the function \( y = 5 \cos x \), the amplitude is \( |5| = 5 \). The absolute value here eliminates any potential negative sign ensuring the amplitude is interpreted correctly as the highest point the wave reaches on a graph, which is a key concept in understanding wave behavior in trigonometry.