Integrating sin2θ: How to Prove <sin2θ>=1/2 and <cos2θ>=1/2?

[gennarakis]

How can <sin²θ>=1/2 and <cos²θ>=1/2

How is the proof made?Integrate sin²θ from -Infinity to +Infinity?

 

[berkeman]

How can <sin²θ>=1/2 and <cos²θ>=1/2

How is the proof made?Integrate sin²θ from -Infinity to +Infinity?

You only need to integrate over one cycle. Every other cycle will be the same, right?

 

[gb7nash]

Average value on what interval?

 

[gennarakis]

I just integrated from 0 to 2Pi changed sin²θ=(1-cos2θ)/2 but the result is Pi and not 1/2...

 

[berkeman]

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...

 

[gennarakis]

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...:)