If theta is considered small in problems

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Discussion Overview

The discussion revolves around the implications of considering the angle theta as small in mathematical problems, particularly focusing on the relationship between theta and its time derivative, theta dot. The scope includes mathematical reasoning and conceptual clarification regarding approximations in trigonometric functions.

Discussion Character

  • Exploratory, Mathematical reasoning, Conceptual clarification

Main Points Raised

  • One participant notes that when theta is small, sin(theta) is approximately equal to theta and questions the behavior of theta dot, the time derivative of theta.
  • Another participant suggests that if theta remains small, the time-derivative of sin(theta) can be approximated by the time-derivative of theta.
  • A later reply points out that since the time derivative of sin is cosine, and cosine approaches one when theta is small, this supports the previous claims.

Areas of Agreement / Disagreement

Participants seem to agree on the approximations related to small theta, but the discussion does not reach a consensus on the implications for theta dot.

Contextual Notes

The discussion does not address potential limitations or assumptions regarding the range of theta or the conditions under which these approximations hold.

M. next
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we know when theta is considered small enough, sin(theta) is approximately equal to theta. Ok, the question is what happens to theta dot(the time derivative of theta)?
 
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If [itex]\theta[/itex] is small all the time, the time-derivative of [itex]sin(\theta)[/itex] is approximately the time-derivative of [itex]\theta[/itex].
 
thanks
 
Another way of seeing the above is that the time derivative of sin is cosine, and when theta is small cosine is approximately equal to one.
 

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