Understanding $\sin\left({\frac{1}{n^2}}\right)$ When $n > 1$

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In summary, since $n>1$ and $\frac{1}{n^2}$ is less than 1, the sine function in the first quadrant will yield a positive result. This is because $n^2>1$, making $\frac{1}{n^2}$ less than 90 degrees or $\frac{\pi}{2}$ radians. Therefore, for $n>1$, $\sin\left(\frac{1}{n^2}\right)>0$.
  • #1
tmt1
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Given than $n > 1$, then $\sin\left({\frac{1}{n^2}}\right) > 0$, but I'm not sure why that is.

I get that a sin function in the first quadrant will yield a positive result, but I'm not sure why it's in the first quadrant in the first place. Would $\frac{1}{n^2}$ be in degrees in this case, in which case, since it's less than 1 it would be in the first quadrant. That makes sense, but I'm not sure how to know that $n$ would be notated in degrees.
 
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  • #2
$0\lt\dfrac{1}{n^2}\lt\dfrac{\pi}{2}$ hence $0\lt\sin\left(\dfrac{1}{n^2}\right)\lt1$ for $n>1$.
 
  • #3
If n> 1 then [tex]n^2> 1[/tex] so [tex]\frac{1}{n^2}< 1[/tex] so [tex]\frac{1}{n^2}[/tex] is certainly less than either 90 (degrees) or [tex]\frac{\pi}{2}[/tex] radians.
 

FAQ: Understanding $\sin\left({\frac{1}{n^2}}\right)$ When $n > 1$

1. What is the pattern of the values of $\sin\left({\frac{1}{n^2}}\right)$ when $n > 1$?

As the value of $n$ increases, the value of $\frac{1}{n^2}$ decreases. This results in an oscillating pattern for $\sin\left({\frac{1}{n^2}}\right)$, where the values get closer to 0 but never reach it.

2. How does the behavior of $\sin\left({\frac{1}{n^2}}\right)$ change as $n$ approaches infinity?

As $n$ approaches infinity, the value of $\frac{1}{n^2}$ approaches 0. This means that the values of $\sin\left({\frac{1}{n^2}}\right)$ will also approach 0, but will never actually reach it.

3. What is the significance of using $\frac{1}{n^2}$ as the input for $\sin$ in this equation?

The use of $\frac{1}{n^2}$ as the input for $\sin$ is a specific choice made to demonstrate the oscillating behavior of the sine function. It allows for a clear visual representation of the graph and helps to understand the concept of limit as $n$ approaches infinity.

4. Can you provide an example of a real-world application of $\sin\left({\frac{1}{n^2}}\right)$ when $n > 1$?

One example of a real-world application is in the study of diffraction patterns. When light passes through a small slit, the resulting diffraction pattern can be described using the function $\sin\left({\frac{1}{n^2}}\right)$, where $n$ represents the number of slits. This can be observed in the interference patterns created by laser beams passing through multiple slits in a diffraction grating.

5. How does the graph of $\sin\left({\frac{1}{n^2}}\right)$ compare to the graph of $\sin(x)$?

The graph of $\sin\left({\frac{1}{n^2}}\right)$ has a similar shape to the graph of $\sin(x)$, but it is compressed along the x-axis. This means that the values of $\sin\left({\frac{1}{n^2}}\right)$ change more rapidly as $x$ (or in this case, $\frac{1}{n^2}$) increases, resulting in a more oscillating pattern. Additionally, the range of values for $\sin\left({\frac{1}{n^2}}\right)$ is limited to between -1 and 1, while the range for $\sin(x)$ is infinite.

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