Problem 26 - Algebra 2 and Trigonometry

Question

Find the phase shift of each function. \(y=\cos (2 x-\pi)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Phase Shift Formula

The phase shift for cosine functions in the form of \(y = a\cos(bx - c) + d\) is given by \(\text{Phase Shift} = \frac{c}{b}\). In this case, the function is \(y = \cos(2x - \pi)\).

Step 2: Extract Values for b and c

From the function \(y = \cos(2x - \pi)\), we identify \(b = 2\) and \(c = \pi\). These values will be used in our phase shift formula.

Step 3: Calculate the Phase Shift

Using the formula \(\text{Phase Shift} = \frac{c}{b}\), substitute the values \(c = \pi\) and \(b = 2\). Thus, the phase shift is \(\frac{\pi}{2}\).

Step 4: Determine Direction of Phase Shift

Since the value of \(\frac{c}{b}\) is positive, the phase shift is to the right by \(\frac{\pi}{2}\) units.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that describes the relationship between the angle in a right-angled triangle and the length of the adjacent side compared to the hypotenuse. It forms part of the family of sine, cosine, and tangent functions, which are pivotal in studying periodic phenomena.
The basic form of the cosine function is denoted as \(y = \cos(x)\). This function generates a smooth curve that repeats every \(2\pi\) units along the x-axis. This repetition makes it incredibly useful for modeling cyclical events, such as sound waves, circular motion, and even day-night cycles.

• The cosine function has a distinctive wave-like shape.
• It reaches its maximum value of 1 and its minimum value of -1.
• The peaks and troughs of the cosine wave indicate symmetrical fluctuations around the x-axis.

For more complex representations, the cosine function can be expressed in a generalized form like \(y = a\cos(bx - c) + d\), where variables control different aspects of the wave:

• \(a\) affects the amplitude (height of the wave).
• \(b\) controls the frequency (how often the wave repeats).
• \(c\) introduces phase shifts (horizontal changes).
• \(d\) adjusts the vertical position of the wave.

Understanding these components helps us manipulate and utilize the cosine function effectively.

Trigonometric Function

Trigonometric functions are essential in mathematics and physics for modeling and solving problems involving angles and distances. They arise from the study of triangles, particularly the right-angled triangle, and have applications far beyond simple geometry.
These functions include sine, cosine, tangent, and their reciprocals—cosecant, secant, and cotangent. Each function relates an angle in a right triangle to a ratio of two side lengths.

• The *sine function* (sin) relates the angle to the ratio of the length of the opposite side to the hypotenuse.
• The *cosine function* (cos), as covered earlier, associates the angle with the adjacent side to hypotenuse ratio.
• The *tangent function* (tan) involves the ratio of the opposite side to the adjacent side.

Trigonometric functions are crucial not just in pure mathematics but also in engineering, astronomy, and various science fields. They help analyze periodic phenomena such as waves and oscillations.
The general properties they share include periodicity, amplitude, and frequency, which allow them to model continuous and repetitive processes efficiently.

• **Periodicity:** Each trigonometric function repeats its values in regular intervals.
• **Amplitude:** This defines the peak value of the function.
• **Frequency:** Determines how often the wave pattern repeats over a specific interval.

Understanding these functions makes it easier to analyze and predict behavior in systems governed by periodic changes.

Phase Shift Formula

Phase shift in trigonometric functions is a horizontal shift along the x-axis. It indicates how far the overall graph of the function has moved to the left or right from its original position.
In the context of the cosine function, the phase shift is determined by the formula \(\text{Phase Shift} = \frac{c}{b}\) from the standard form \(y = a\cos(bx - c) + d\). The components \(b\) and \(c\) specifically influence how much and in what direction the graph shifts:

• \(c\): Relates directly to the shift amount. A positive \(c\) means a shift to the left, whereas a negative \(c\) indicates a shift to the right.
• \(b\): Inversely affects the width of the period. The larger the \(b\), the more compressed the graph, hence influencing how the shift manifests across the x-axis.

For instance, in the equation \(y = \cos(2x - \pi)\), we set \(b = 2\) and \(c = \pi\). Plugging these into our phase shift formula, \(\text{Phase Shift} = \frac{c}{b} = \frac{\pi}{2}\), we conclude that the shift is \(\frac{\pi}{2}\) units to the right because the resulting fraction is positive.
This phase shift can be easily noticed if you graph the cosine function: the entire wave moves by \(\frac{\pi}{2}\) along the x-axis. Understanding and applying phase shifts is crucial when precisely modeling periodic signals like alternating currents and sound waves.