What is the average for cos2(x)?

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SUMMARY

The average value of cos²(x) is definitively 1/2, as derived from the identity cos²(x) = (1 + cos(2x))/2. Integration over a full period confirms this result, yielding an average of 0.5. Additionally, using the Pythagorean identity cos²(x) + sin²(x) = 1, both terms share the same average, reinforcing that the average for cos²(x) is indeed 1/2.

PREREQUISITES
  • Understanding of trigonometric identities, specifically cos²(x) and sin²(x).
  • Basic knowledge of integration techniques in calculus.
  • Familiarity with the concept of average value in periodic functions.
  • Knowledge of the cosine function and its properties over intervals.
NEXT STEPS
  • Study the derivation of trigonometric identities, focusing on cos²(x) and sin²(x).
  • Learn integration techniques for periodic functions, particularly over full cycles.
  • Explore the application of the average value theorem in trigonometry.
  • Investigate the properties of the cosine function and its graphical representation.
USEFUL FOR

Students of mathematics, particularly those studying calculus and trigonometry, as well as educators seeking to explain the average values of trigonometric functions.

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What is the expectation value of cosine squared, namely <cos^2(t)>?
 
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cos2x = (1+cos(2x))/2

Integration leads to 0,5 (averaged over the lenght)
 
Another approach: cos2(x) + sin2(x) = 1. Both terms have the same average, so the average for cos2(x) = 1/2.
 

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