Understanding the Transition to Solve for t in an Oscillation Function

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SUMMARY

The discussion focuses on solving for time (t) in the oscillation function x = A cos(wt). The correct formula derived is t = (arccos(x/A) / (2π)) * T, where T represents the period of the oscillation. The transition involves dividing the equation by A and applying the inverse cosine function, arccos. The relationship between angular frequency (w) and period (T) is established through the equation w = 2π/T.

PREREQUISITES
  • Understanding of trigonometric functions, specifically cosine and its inverse, arccos.
  • Familiarity with oscillation concepts, including amplitude (A) and angular frequency (w).
  • Knowledge of the relationship between angular frequency and period (T).
  • Basic algebraic manipulation skills to isolate variables in equations.
NEXT STEPS
  • Study the properties of inverse trigonometric functions, particularly arccos.
  • Learn about the derivation of oscillation equations in physics, focusing on harmonic motion.
  • Explore the relationship between angular frequency and period in greater detail.
  • Practice solving various oscillation problems to reinforce understanding of time isolation in trigonometric functions.
USEFUL FOR

Students studying physics or mathematics, particularly those focusing on oscillatory motion and trigonometric functions. This discussion is beneficial for anyone looking to deepen their understanding of solving equations involving cosine functions.

Mynona
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I have a cosine function, namely the function for oscillation x=A cos(wt).

I want to separate the t out here, so I can solve for it. My teacher gave the the answer to be t= (arccos (x/a)/2pi)*T, but I can't quite see where he came up with that. Would anyone be as kind as to give me a more elaborate explanation of how this transition was made.


(2pi and T comes from w=2pi/T).

Sorry for bad English, not a native English speaker obviously :)
 
Mathematics news on Phys.org
The inverse function of the cosine is the arc cosine. If a function f(x) has an inverse f^{-1}(x) then f^{-1}(f(x))=f(f^{-1}(x))=x.

Divide the equation on both sides by A then take the arccos on both sides. Can you calculate \arccos(\cos(\omega t))now?
 

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