Exploring Orthogonality: A Homework Challenge

In summary, the conversation is about finding the integral of d(theta) sin^2 k(theta) cos^2 k(theta) from 0 to 2pi using the trig identity sin(2x)=2*sin(x)*cos(x). The integral simplifies to 1/8 times the difference between 1 and sin(4k*theta), which becomes pi/4 if k is an integer and varies for non-integer values of k.
  • #1
bigplanet401
104
0

Homework Statement


What is

[tex]
\int_0^{2 \pi} \; d\theta \sin^2 k\theta \cos^2 k\theta \; ?
[/tex]

Homework Equations


Orthogonality of sines and cosines?

The Attempt at a Solution


I tried substitution and didn't get anywhere. Yeah, that's about it.
 
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  • #2
Use the trig identity sin(2x)=2*sin(x)*cos(x) for a start.
 
  • #3
Whoa, I completely missed that. Using that identity, the integral becomes

[tex]
\begin{align*}
& \int_0^{2\pi} d\theta \; \frac{1}{4} \sin^2 2k\theta\\
&= \int_0^{2\pi} d\theta \; \frac{1}{8} (1 - \sin 4 k \theta )\\
&= \frac{\pi}{4} \, ,
\end{align*}
[/tex]
right?

The answer just seems too simple--like there should be some k's around or something.
 
  • #4
You mean 1-cos(4k*theta), I hope. If k is an integer then there are no k's left around. If it isn't there are.
 

1. What is "Exploring Orthogonality: A Homework Challenge"?

"Exploring Orthogonality: A Homework Challenge" is a scientific exercise that involves analyzing a dataset to identify and quantify the level of orthogonality between variables. It is often used in the field of statistics and data analysis to assess the relationships between different variables.

2. Why is orthogonality important in scientific research?

Orthogonality is important in scientific research because it allows us to understand the unique contribution of each variable to a given phenomenon. It helps us to identify which variables are truly independent and have a significant impact on the outcome being studied.

3. How is orthogonality calculated in "Exploring Orthogonality: A Homework Challenge"?

Orthogonality is typically calculated using a statistical measure called the correlation coefficient. This measures the strength and direction of the linear relationship between two variables, and a value of zero indicates complete orthogonality.

4. Can orthogonality be applied to non-numerical data?

Yes, orthogonality can also be applied to non-numerical data. In this case, we would use different statistical measures such as chi-square tests or contingency tables to assess the relationships between categorical variables.

5. How can "Exploring Orthogonality: A Homework Challenge" benefit scientific research?

"Exploring Orthogonality: A Homework Challenge" can benefit scientific research by providing a standardized and objective method for assessing the relationships between variables. It can help researchers to identify and control for confounding factors, leading to more accurate and reliable results.

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