Identifying Phase Shift In Trigonometric Functions A Step By Step Guide

Trigonometric functions are fundamental in mathematics, physics, and engineering, and understanding their transformations is crucial for various applications. One such transformation is the phase shift, which involves the horizontal translation of a trigonometric function's graph. In this article, we will delve into the concept of phase shift, explore how to identify it in trigonometric equations, and provide a step-by-step solution to the given problem.

The phase shift of a trigonometric function represents the horizontal displacement of its graph from its original position. It essentially indicates how much the function has been shifted to the left or right along the x-axis. In the general form of a sinusoidal function, such as $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$, the phase shift is determined by the parameter C. A positive value of C indicates a shift to the right, while a negative value indicates a shift to the left.

Key Concepts of Phase Shift

Before diving into the problem, let's solidify our understanding of the key concepts related to phase shifts:

1. General Form: The general form of a sinusoidal function is given by:

   $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$

   where:

   • A represents the amplitude.
   • B affects the period of the function.
   • C represents the phase shift.
   • D represents the vertical shift.

2. Phase Shift Calculation: The phase shift is calculated as the value of C. It's crucial to express the function in the form $y = A \sin(B(x - C))$ or $y = A \cos(B(x - C))$ to accurately identify C.

3. Direction of Shift: A positive value of C indicates a shift to the right, while a negative value indicates a shift to the left.

4. Period and Phase Shift: The parameter B affects the period of the function. The period is given by $T = \frac{2\pi}{|B|}$. The phase shift is relative to the period, so a larger B value will compress the graph horizontally, affecting the perceived shift.

Problem Analysis

The problem asks us to identify the function that has a phase shift of $\frac{\pi}{2}$ to the right. This means we are looking for a function where C is equal to $\frac{\pi}{2}$ when the function is expressed in the form $y = A \sin(B(x - C))$. Let's analyze each option:

Option A: $y = 2 \sin(x + \frac{\pi}{2})$

To determine the phase shift, we need to rewrite the equation in the form $y = A \sin(B(x - C))$.

$y = 2 \sin(x + \frac{\pi}{2})$ can be rewritten as $y = 2 \sin(1(x - (-\frac{\pi}{2})))).

Here, $A = 2$, $B = 1$, and $C = -\frac{\pi}{2}$. Since C is negative, this function has a phase shift of $\frac{\pi}{2}$ to the left, not the right. Therefore, Option A is incorrect.

Option B: $y = 2 \sin(\frac{1}{2}x + \pi)$

To find the phase shift, we need to factor out the coefficient of x from the argument of the sine function:

$y = 2 \sin(\frac{1}{2}x + \pi) = 2 \sin(\frac{1}{2}(x + 2\pi))$

Now, we can rewrite it in the form $y = A \sin(B(x - C))$:

$y = 2 \sin(\frac{1}{2}(x - (-2\pi)))$

Here, $A = 2$, $B = \frac{1}{2}$, and $C = -2\pi$. Since C is negative, this function has a phase shift of $2\pi$ to the left, not the right. Therefore, Option B is incorrect.

Option C: $y = 2 \sin(x - \pi)$

This equation is already in the form $y = A \sin(B(x - C))$:

$y = 2 \sin(1(x - \pi))$

Here, $A = 2$, $B = 1$, and $C = \pi$. Since C is positive, this function has a phase shift of $\pi$ to the right. This is not the phase shift we are looking for, so Option C is incorrect.

Option D: $y = 2 \sin(2x - \pi)$

To determine the phase shift, we need to factor out the coefficient of x from the argument of the sine function:

$y = 2 \sin(2x - \pi) = 2 \sin(2(x - \frac{\pi}{2}))$

Now, we can see that the equation is in the form $y = A \sin(B(x - C))$:

$y = 2 \sin(2(x - \frac{\pi}{2}))$

Here, $A = 2$, $B = 2$, and $C = \frac{\pi}{2}$. Since C is positive, this function has a phase shift of $\frac{\pi}{2}$ to the right. Therefore, Option D is the correct answer.

Step-by-Step Solution

To summarize, here's a step-by-step approach to solving this type of problem:

1. Identify the General Form: Recognize the general form of a sinusoidal function: $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$.
2. Rewrite the Equation: Rewrite the given equation in the general form by factoring out the coefficient of x from the argument of the trigonometric function.
3. Determine the Phase Shift: Identify the value of C in the rewritten equation. This value represents the phase shift.
4. Determine the Direction: If C is positive, the phase shift is to the right. If C is negative, the phase shift is to the left.
5. Compare with the Problem Statement: Compare the phase shift and direction with the problem statement to select the correct answer.

Common Mistakes to Avoid

• Forgetting to Factor: A common mistake is forgetting to factor out the coefficient of x when determining the phase shift. For example, in the equation $y = 2 \sin(2x - \pi)$, it's crucial to factor out the 2 to get $y = 2 \sin(2(x - \frac{\pi}{2}))$.
• Incorrect Sign: Ensure you pay close attention to the sign of C. A positive C indicates a shift to the right, while a negative C indicates a shift to the left.
• Misinterpreting the Shift: The phase shift represents the horizontal displacement of the graph. Make sure you understand the direction (left or right) and the magnitude of the shift.

Conclusion

In conclusion, the function with a phase shift of $\frac{\pi}{2}$ to the right is Option D: $y = 2 \sin(2x - \pi)$. By understanding the general form of sinusoidal functions and following the step-by-step approach, you can confidently solve problems involving phase shifts. Remember to factor out the coefficient of x, pay attention to the sign of C, and carefully interpret the shift.

Mastering the concept of phase shifts is essential for understanding more complex trigonometric transformations and their applications in various fields. By practicing and applying these concepts, you can strengthen your understanding of trigonometric functions and their behavior.