- #1
gennarakis
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How can <sin2θ>=1/2 and <cos2θ>=1/2
How is the proof made?Integrate sin2θ from -Infinity to +Infinity?
How is the proof made?Integrate sin2θ from -Infinity to +Infinity?
Integrating sin2θ: How to Prove <sin2θ>=1/2 and <cos2θ>=1/2?
- Thread starter gennarakis
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In summary, the conversation discusses finding the average value of sin2θ and cos2θ when they equal 1/2. The proof involves integrating over one cycle and dividing by 2π to get the average over the interval. The result is Pi, and not 1/2 as initially thought. The mistake was not dividing by 2π to get the correct average value.
- #2
berkeman
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gennarakis said:
You only need to integrate over one cycle. Every other cycle will be the same, right?
- #3
gb7nash
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Average value on what interval?
- #4
gennarakis
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I just integrated from 0 to 2Pi changed sin2θ=(1-cos2θ)/2 but the result is Pi and not 1/2...
- #5
berkeman
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gennarakis said:I just integrated from 0 to 2Pi changed sin2θ=(1-cos2θ)/2 but the result is Pi and not 1/2...
You forgot to divide by 2*PI to get the average over the interval...
- #6
gennarakis
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Sorry as it seems I didn't use the exact average rule... http://upload.wikimedia.org/math/2/4/7/247a435e08fbaf845d6501ac1ca740a7.png
Thanks!It was my first answer on PF and I have to say it was fast and easy...:)
Thanks!It was my first answer on PF and I have to say it was fast and easy...:)
Related to Integrating sin2θ: How to Prove <sin2θ>=1/2 and <cos2θ>=1/2?
1. What is the average value of sin^2(θ)?
The average value of sin^2(θ) is 1/2. This means that if you were to take multiple measurements of sin^2(θ) at different values of θ, the average of all those values would approach 1/2.
2. How is the average value of sin^2(θ) calculated?
The average value of sin^2(θ) is calculated by integrating sin^2(θ) over the range of θ and dividing by the range of θ. This can be represented by the formula: average value = (1/π) x ∫sin^2(θ) dθ.
3. Why is the average value of sin^2(θ) important?
The average value of sin^2(θ) is important because it is a fundamental concept in trigonometry and calculus. It is also used in many applications, such as in signal processing and Fourier analysis.
4. What is the physical significance of the average value of sin^2(θ)?
The average value of sin^2(θ) has physical significance in the context of waves and vibrations. It represents the average amplitude of a wave or the average energy of a vibrating system over a given period of time.
5. Can the average value of sin^2(θ) be greater than 1?
No, the average value of sin^2(θ) cannot be greater than 1. This is because sin^2(θ) is always less than or equal to 1 for any value of θ. Therefore, the average value of sin^2(θ) will also be less than or equal to 1.
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