Solve "Circles and Sectors: 3θ=2(π−sinθ)

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SUMMARY

The discussion focuses on solving the equation derived from the geometry of a semicircle, specifically the relationship between the angle POB (θ) and the areas of the sector and shaded segment. The key equation established is 3θ = 2(π - sinθ), which arises from the condition that the area of sector POB equals twice the area of the shaded segment. The problem involves using the area formulas for a circle and segment, specifically Area of circle = (1/2)r²θ and Area of segment = (1/2)r²(θ - sinθ). The challenge lies in correctly applying these formulas to derive the required relationship.

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  • Understanding of semicircles and their properties
  • Knowledge of trigonometric functions and their applications
  • Familiarity with area formulas for circles and segments
  • Basic algebraic manipulation skills
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  • Learn about the properties of semicircles and their geometric implications
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Students studying geometry, particularly those tackling problems involving semicircles and trigonometric relationships, as well as educators looking for examples of geometric problem-solving techniques.

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Homework Statement


The diagram shows a semicircle APB on AB as diameter. The midpoint of AB is O. The point P on the semicircle is such that the area of the sector POB is equal to twice the area of the shade segment. Given that angle POB is [tex]\theta[/tex] radians, show that

3[tex]\theta[/tex] = 2([tex]\pi[/tex]-sin[tex]\theta[/tex])​


Homework Equations





The Attempt at a Solution


using formula
Area of circle = [tex]\frac{1}{2}[/tex]r2[tex]\theta[/tex]
and
Area of segment = [tex]\frac{1}{2}[/tex]r2 ([tex]\theta[/tex] - sin [tex]\theta[/tex] )
heres the problems
from the picture http://img130.imageshack.us/img130/1790/001tz.jpg
questions 4
the the angle of the segment is [tex]\pi[/tex]-[tex]\theta[/tex]
there I am clueless even i inserted the info i have
what i really get is
[tex]\theta[/tex]=2[[tex]\pi[/tex]-[tex]\theta[/tex]-sin([tex]\pi[/tex]-[tex]\theta[/tex])]​
of course we can't use formula blindly so anyone can help me there
 
Last edited by a moderator:
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If sector POA contains the shaded segment and triangle POA, how do you find the area of the shaded region?
 

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