Equivalent Trigonometric Functions For Y=3cos(2(x+π/2))-2

Jul 13, 2025 by Admin 58 views

In the realm of trigonometry, exploring the equivalence of functions is a fascinating endeavor. This article delves into the process of identifying trigonometric functions that are identical to a given function. Our focus is on the function y=3cos(2(x+π/2))-2, and we aim to determine which of the provided options accurately represent the same function. This exploration involves understanding the transformations and identities that govern trigonometric functions, allowing us to manipulate and rewrite expressions to reveal their underlying relationships. By meticulously analyzing the amplitude, period, phase shift, and vertical shift of each function, we can unveil the equivalent form that perfectly mirrors the behavior of the original function. Let's embark on this trigonometric journey, where we will dissect each option and unveil the hidden connections between them.

Understanding the Original Function: y=3cos(2(x+π/2))-2

To begin our quest, let's first dissect the original function, y=3cos(2(x+π/2))-2, to understand its key characteristics. This function is a cosine function that has undergone several transformations. The amplitude, which determines the vertical stretch, is 3. This means the function's maximum and minimum values will be 3 units away from its midline. The argument of the cosine function, 2(x+π/2), reveals the period and phase shift. The period is determined by the coefficient of x, which is 2. The period is calculated as 2π divided by the coefficient of x, giving us a period of π. This indicates that the function completes one full cycle in an interval of π units along the x-axis.

Furthermore, the term (x+π/2) indicates a phase shift, a horizontal translation of the function. Specifically, the function is shifted π/2 units to the left. This means that the graph of the cosine function starts its cycle π/2 units earlier than the standard cosine function. Finally, the constant term -2 represents a vertical shift, moving the entire function down by 2 units. This shift affects the midline of the function, which is the horizontal line about which the function oscillates. The midline for this function is y=-2.

In summary, the function y=3cos(2(x+π/2))-2 is a cosine function with an amplitude of 3, a period of π, a phase shift of π/2 units to the left, and a vertical shift of 2 units downward. This comprehensive understanding of the function's attributes sets the stage for our analysis of the options, where we will carefully compare their characteristics to those of the original function. By meticulously examining each option's amplitude, period, phase shift, and vertical shift, we will identify the function that perfectly aligns with the behavior of y=3cos(2(x+π/2))-2. This comparative analysis will involve utilizing trigonometric identities and transformations to rewrite the options in forms that are directly comparable to the original function, ultimately leading us to the equivalent expression.

Option 1: y=3sin(2(x+π/4))-2

The first option we'll examine is y=3sin(2(x+π/4))-2. This is a sine function, and to determine if it's equivalent to our original cosine function, we need to delve into trigonometric identities and transformations. The amplitude of this sine function is 3, which matches the amplitude of the original cosine function. The period is determined by the coefficient of x within the sine function's argument, which is 2. Therefore, the period is 2π/2 = π, again matching the original function. The phase shift is given by the term (x+π/4), indicating a shift of π/4 units to the left. The vertical shift is -2, which also matches the original function.

However, the crucial difference lies in the fact that this is a sine function, while our original function is a cosine function. We can use the trigonometric identity cos(θ) = sin(θ + π/2) to relate cosine and sine functions. Applying this identity to our original function, we get:

y = 3cos(2(x+π/2))-2 = 3sin(2(x+π/2) + π/2) - 2

Simplifying the argument of the sine function:

2(x+π/2) + π/2 = 2x + π + π/2 = 2x + 3π/2 = 2(x + 3π/4)

Therefore, the original function can be rewritten as:

y = 3sin(2(x + 3π/4)) - 2

Now, comparing this to the first option, y=3sin(2(x+π/4))-2, we see that the phase shifts are different. The original function, when converted to sine, has a phase shift of 3π/4 to the left, while the first option has a phase shift of π/4 to the left. Therefore, this option is not equivalent to the original function. This analysis highlights the importance of carefully considering phase shifts when comparing trigonometric functions. Even if the amplitudes, periods, and vertical shifts match, a difference in phase shift can lead to entirely different function behaviors. The use of trigonometric identities is crucial in bridging the gap between sine and cosine functions, allowing for a direct comparison of their properties.

Option 2: y=-3sin(2(x+π/4))-2

Moving on to the second option, we have y=-3sin(2(x+π/4))-2. This function also involves a sine term, but with a negative amplitude. The negative sign in front of the amplitude indicates a reflection across the x-axis. The amplitude itself is the absolute value of the coefficient, which is |-3| = 3, matching the original function's amplitude. The period is determined by the coefficient of x, which is 2, resulting in a period of π, consistent with the original function. The phase shift is π/4 units to the left, as indicated by the term (x+π/4). The vertical shift is -2, matching the original function.

To compare this option with the original function, we again use the identity cos(θ) = sin(θ + π/2) to rewrite the original function in terms of sine:

y = 3cos(2(x+π/2))-2 = 3sin(2(x+π/2) + π/2) - 2 = 3sin(2(x + 3π/4)) - 2

Now, we need to account for the negative amplitude in the second option. We can use the identity sin(θ + π) = -sin(θ) to introduce a negative sign. Applying this to the second option, we can rewrite it as:

y = -3sin(2(x+π/4))-2 = 3(-sin(2(x+π/4)))-2 = 3(sin(2(x+π/4) + π)) - 2

Simplifying the argument of the sine function:

2(x+π/4) + π = 2x + π/2 + π = 2x + 3π/2 = 2(x + 3π/4)

Therefore, the second option can be rewritten as:

y = 3sin(2(x + 3π/4)) - 2

Comparing this to the rewritten original function, y = 3sin(2(x + 3π/4)) - 2, we find that they are identical. This confirms that the second option, y=-3sin(2(x+π/4))-2, is indeed equivalent to the original function. This equivalence demonstrates the interplay between amplitude, phase shift, and reflections in trigonometric functions. By carefully applying trigonometric identities and transformations, we can unveil the hidden relationships between seemingly different expressions.

Option 3: y=3cos(2x+π)-2

Finally, let's consider the third option: y=3cos(2x+π)-2. This function is expressed in terms of cosine, making it directly comparable to our original function. The amplitude is 3, which matches the original. The period can be determined from the coefficient of x, which is 2, resulting in a period of π, consistent with the original function. To analyze the phase shift, we need to rewrite the argument in the form 2(x + C), where C represents the phase shift. We can rewrite the argument as follows:

2x + π = 2(x + π/2)

This reveals a phase shift of π/2 units to the left, which matches the original function. The vertical shift is -2, also consistent with the original function.

Comparing this option, y=3cos(2(x+π/2))-2, directly with the original function, y=3cos(2(x+π/2))-2, we see that they are identical. Therefore, this option is also equivalent to the original function. This demonstrates that multiple equivalent forms can exist for a single trigonometric function, highlighting the flexibility and interconnectedness of trigonometric expressions.

Conclusion

In conclusion, after a thorough analysis of all options, we have identified two functions that are equivalent to the original function y=3cos(2(x+π/2))-2. These equivalent functions are:

y=-3sin(2(x+π/4))-2
y=3cos(2x+π)-2

This exploration highlights the importance of understanding trigonometric identities and transformations in identifying equivalent functions. By carefully analyzing the amplitude, period, phase shift, and vertical shift of each function, and by utilizing identities to bridge the gap between sine and cosine functions, we can unveil the underlying relationships between seemingly different expressions. This ability to manipulate and rewrite trigonometric functions is a crucial skill in various fields, including mathematics, physics, and engineering, where trigonometric functions play a fundamental role in modeling periodic phenomena.

The process of finding equivalent trigonometric functions involves a deep understanding of their properties and the transformations they undergo. By mastering these concepts, we gain the ability to navigate the intricate world of trigonometric expressions and reveal their hidden connections.