How Can the Maclaurin Series Validate a Trigonometric Inequality?

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Homework Help Overview

The problem involves validating a trigonometric inequality using the Maclaurin series for the cosine function. The specific inequality to be shown is 1 - t²/2 ≤ cos(t) ≤ 1 for the interval 0 ≤ t ≤ 1.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express uncertainty about how to relate the variable t to cos(t) and question the bounds of the inequality, particularly whether the interval should be 0 ≤ t ≤ π/2 instead of 0 ≤ t ≤ 1.

Discussion Status

The discussion is ongoing, with participants exploring the relevance of the Maclaurin series for cos(t) and questioning the problem's constraints. Some guidance is offered regarding the context of the calculus course, but no consensus has been reached on the approach to take.

Contextual Notes

There is a noted confusion regarding the appropriate interval for t, as participants question whether the original problem statement is accurate. Additionally, the relevance of the Maclaurin series is brought into the discussion without a clear direction on its application.

Clara Chung
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Homework Statement


show that 1-t^2/2 <=cos(t) <=1 for 0<=t<=1

Homework Equations


Trigonometry knowledge

The Attempt at a Solution


I don't know how to relate t with cos(t), and I also try to find out cos(1), but there is no result, so how can I start with this problem.
 
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Clara Chung said:

Homework Statement


show that 1-t^2/2 <=cos(t) <=1 for 0<=t<=1

Homework Equations


Trigonometry knowledge

The Attempt at a Solution


I don't know how to relate t with cos(t), and I also try to find out cos(1), but there is no result, so how can I start with this problem.
Are you sure that it's 0 ≤ t ≤ 1 , and not 0 ≤ t ≤ π/2 ?
 
Yes, this is what the question said. How can you solve for 0 <= t <= pi/2 ?
 
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Clara Chung said:
show that 1-t^2/2 <=cos(t) <=1 for 0<=t<=1
Have you know about Maclaurin series yet, and in particular, the Maclaurin series for cos(t)?

From another question you posted, you are taking a calculus class. If so, both questions should have been posted in the Calculus & Beyond section, not the Precalculus section.
 

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