Problem 20 - Algebra 2 and Trigonometry

Question

Find the phase shift of each function. \(y=\cos \left(x-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

The phase shift is \( \frac{\pi}{2} \) to the right.

Step 1: Identify the General Form

The general form of a cosine function with a phase shift is given by \( y = \cos(bx - c) \). Our task is to identify the phase shift from this formula. Compare the given function to this form.

Step 2: Extract the Components

From \( y = \cos(x - \frac{\pi}{2}) \), notice that \( b = 1 \) and \( c = \frac{\pi}{2} \). These help in determining the phase shift using the formula \( \frac{c}{b} \).

Step 3: Calculate the Phase Shift

Use the formula for phase shift \( \text{Phase Shift} = \frac{c}{b} \). Thus, \( \frac{\pi}{2} \) divided by \( 1 \) gives the phase shift as \( \frac{\pi}{2} \).

Step 4: Determine the Direction

Since the phase shift expression is \( x - \frac{\pi}{2} \), it indicates a shift to the right. Generally, if \( c > 0 \), the shift is to the right. Therefore, the phase shift is \( \frac{\pi}{2} \) to the right.

Key Concepts

Cosine Function

The cosine function, a core concept in trigonometry, is one of the most well-known trigonometric functions. It is often written as \( \cos(x) \). The behavior of this function is periodic, meaning it repeats its values at regular intervals. When graphed, the cosine function forms a wave-like pattern known as a cosine wave. The basic properties of this wave include:

• Amplitude: This is essentially the height of the wave, defined as the maximum distance from the midline to the peak of the wave.
• Period: For the standard cosine function, the period is \( 2\pi \), which means the wave pattern repeats every \( 2\pi \) units.
• Frequency: This tells us how often the wave repeats within a unit interval, calculated as \( \frac{1}{\text{Period}} \).
• Midline: It is the horizontal line around which the wave oscillates, typically at \( y = 0 \) for the basic cosine function.

Understanding these fundamental aspects allows for transformations like shifting, stretching, and reflecting the wave to model various phenomena in real-world contexts.

Trigonometric Functions

Trigonometric functions are functions that relate the angles of a triangle to the lengths of its sides. Beyond angle measurements, they can describe many cyclical phenomena. The primary trigonometric functions are sine, cosine, and tangent.

• Sine Function: Like cosine, it is periodic and primarily used to define the vertical component of a circular motion.
• Cosine Function: Focuses on the horizontal component, and is crucial in defining angles and sides in right-angled triangles.
• Tangent Function: Represents the ratio of the sine to cosine of an angle, often used for finding slopes and inclines.

Each of these functions has a reciprocal version: cosecant, secant, and cotangent. They extend the functionality of trigonometric expressions and are often applied in advanced mathematical contexts. They are fundamental in fields like physics, engineering, and even finance, for understanding periodic phenomena.

General Form of Cosine Function

The general form of the cosine function is essential for understanding how alterations in its standard form affect its graph. Typically, the function is expressed as: \( y = a \cos(bx - c) + d \). This notation allows for transformations such as:

• Amplitude \( a \): Modifies the height of the wave; larger values stretch the wave vertically.
• Frequency and Period \( b \): Alters the number of cycles within a specific interval, with the period calculated as \( \frac{2\pi}{b} \).
• Phase Shift \( c \): Moves the graph horizontally, calculated by \( \frac{c}{b} \).
• Vertical Shift \( d \): Moves the graph up or down on the y-axis.

Specifically, for phase shift, this parameter represents how much the entire graph shifts horizontally from its original position. In the equation \( y = \cos(x - \frac{\pi}{2}) \), identifying the phase shift as \( \frac{\pi}{2} \) reflects a rightward movement. This understanding supports modeling various periodic behaviors in practical applications like sound waves and seasonal cycles.