Problem 170 - Calculus Volume 1

Question

Find a. the amplitude, b. the period, and c. the phase shift with direction for each function. \(y=2 \cos \left(x-\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

a. Amplitude: 2 b. Period: \(2\pi\) c. Phase shift: \(\frac{\pi}{3}\) to the right

Step 1: Identifying the standard form

The function is given as \( y = 2 \cos \left( x - \frac{\pi}{3} \right) \). In general, a cosine function is of the form \( y = a \cos(b(x - c)) + d \). Here, \( a = 2 \), \( b = 1 \), \( c = \frac{\pi}{3} \), and \( d = 0 \).

Step 2: Finding the amplitude

The amplitude of a cosine function is the absolute value of \( a \). Here, \( a = 2 \), so the amplitude is \( |2| = 2 \).

Step 3: Calculating the period

The period of a cosine function is given by \( \frac{2\pi}{b} \). Since \( b = 1 \) in this function, the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Determining the phase shift

The phase shift of a cosine function is determined by the value \( c \) in the expression \( b(x - c) \). Here, \( c = \frac{\pi}{3} \). Because it is subtracted, the phase shift is \( \frac{\pi}{3} \) to the right.

Key Concepts

Cosine Function

The cosine function is one of the most fundamental functions in trigonometry. It is often denoted as \( \cos(x) \) and represents the adjacent side over hypotenuse ratio in a right triangle. The cosine function is periodic and oscillates between 1 and -1, repeating its values at regular intervals known as periods. The general form of the cosine function is given as:

• \( y = a \cos(b(x - c)) + d \)

Here:

• \( a \) is the amplitude, controlling the vertical stretch or compression.
• \( b \) affects the period, or the horizontal stretch.
• \( c \) determines the phase shift, or the horizontal translation.
• \( d \) adjusts the vertical shift, moving the function up or down.

This function plays a crucial role in modeling waves and oscillations in real-world scenarios, such as sound waves and light waves.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are essential in understanding relationships in right triangles and modeling periodic phenomena. These functions are defined based on the unit circle, a circle with a radius of 1, centered at the origin of the coordinate plane. When dealing with trigonometric functions:

• Sine, denoted as \( \sin(x) \), is the y-coordinate of the point on the unit circle.
• Cosine, denoted as \( \cos(x) \), is the x-coordinate.
• Tangent, \( \tan(x) \), is the ratio of sine to cosine, or \( \frac{\sin(x)}{\cos(x)} \).

These functions are periodic, meaning they repeat their values in a consistent pattern. Cosine and sine have periods of \( 2\pi \), corresponding to one complete revolution around the unit circle. Trigonometric functions are pivotal in a wide range of applications including physics, engineering, and computer graphics.

Phase Shift Calculation

Phase shift in trigonometric functions refers to the horizontal movement along the x-axis. For the cosine function \( y = a \cos(b(x - c)) + d \), the phase shift is determined by the value \( c \). When \( x \) is replaced by \( (x - c) \), the graph of the cosine function shifts to the right if \( c \) is positive and to the left if \( c \) is negative. This is calculated as:

• Phase shift \( = c \) (for the form \( b(x - c) \))

For example, in the function \( y = 2 \cos(x - \frac{\pi}{3}) \), the phase shift is \( \frac{\pi}{3} \) to the right. Phase shifts are particularly useful in modeling scenarios where the starting point of a periodic phenomenon does not align with the origin, requiring a horizontal adjustment.

Amplitude of a Function

Amplitude is a measure of how much a periodic function varies from its central value. In cosine or sine functions, it represents the height of peaks and the depth of troughs from the midline (the horizontal line around which the function oscillates). The amplitude of the function \( y = a \cos(b(x - c)) + d \) is \( |a| \). It is always expressed as a positive value:

• Amplitude \( = |a| \)

For example, with the cosine function \( y = 2 \cos(x - \frac{\pi}{3}) \), the amplitude is \( |2| = 2 \). Knowing the amplitude is essential, especially in contexts like sound waves, to understand the loudness or intensity, and in physics, to assess the extent of oscillations.