How to evaluate -arctan(cosx) from ∏/2 to ∏

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Homework Help Overview

The discussion revolves around evaluating the integral of sin(x)/(1+cos²(x)) from π/2 to π, with a focus on the resulting expression -arctan(cos(x)). Participants are exploring the relationship between trigonometric functions and their inverses in the context of definite integrals.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the integration process and the application of the Fundamental Theorem of Calculus. Questions arise about how to evaluate the expression -arctan(cos(x)) at the bounds π/2 and π, including considerations of using the unit circle or triangles to find values.

Discussion Status

Some participants have provided values for cos(π) and cos(π/2), and there is a suggestion to find the corresponding arctan values. The discussion is progressing with participants verifying their understanding of trigonometric values and their implications for the integral evaluation.

Contextual Notes

Participants are navigating the relationship between trigonometric functions and their inverses, with an emphasis on understanding how to apply these concepts within the framework of definite integrals. There is an acknowledgment of potential confusion regarding the evaluation process.

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Homework Statement


The direction is the evaluate the integral, this isn't really a calculus issue, it's more of a trig issue.
∫sinx(dx)/(1+cos^2x) from ∏/2 to ∏

Homework Equations


(1/a)arctan(u/a)+c

The Attempt at a Solution


I did all the integrating and ended up at
-arctan(cosx) from ∏/2 to ∏
this is where I'm stuck, I know I'm suppose to use the fundamental theorum of Calculus but I don't know what to do once I plug in ∏/2 and ∏. How do I generate values out of this? Do I draw a triangle? Do I use the unit circle? If so, how would I use it
 
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mathnoobie said:

Homework Statement


The direction is the evaluate the integral, this isn't really a calculus issue, it's more of a trig issue.
∫sinx(dx)/(1+cos^2x) from ∏/2 to ∏

Homework Equations


(1/a)arctan(u/a)+c



The Attempt at a Solution


I did all the integrating and ended up at
-arctan(cosx) from ∏/2 to ∏
this is where I'm stuck, I know I'm suppose to use the fundamental theorum of Calculus but I don't know what to do once I plug in ∏/2 and ∏. How do I generate values out of this? Do I draw a triangle? Do I use the unit circle? If so, how would I use it

This seems pretty straightforward.
-arctan(cos([itex]\pi[/itex])) - (-arctan(cos([itex]\pi[/itex]/2)))

What is cos([itex]\pi[/itex])? cos([itex]\pi[/itex]/2)?
 
Cos ∏ is -1
Cos ∏/2 is 0 I believe
so then I would take the Arctan(-1)-Arctan(0)
So I would just find on the unit circle where tangent equals -1 and 0, then subtract?
 

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