- #1
DuncanM
- 98
- 2
Consider the following:
On a circle of radius 1, two points are marked: P1 and P2.
Two lines are drawn from the center of the circle:
one from the center to P1,
the other from the center to P2.
The angle between these two lines is [itex]\theta[/itex].
One more line is drawn: from P1 directly to P2. In other words, this third line is a chord on this circle.
For the special condition that the length of this chord equals the angle, find a simple expression.
i.e. – find a simple expression for [itex]\theta[/itex] given the special condition that chord length = [itex]\theta[/itex] = angle = [itex]\theta[/itex]
- - -
So far, all the expressions that I have worked out mix terms of [itex]\theta[/itex] and either sin([itex]\theta[/itex]) or cos([itex]\theta[/itex]); I have not been able to find an expression simply in terms of [itex]\theta[/itex], sin([itex]\theta[/itex]), or cos([itex]\theta[/itex]).
For example, following is one of my approaches:
Bisect the angle [itex]\theta[/itex], which also divides the chord in half.
The chord length is [itex]\theta[/itex].
But this value is also 2 sin([itex]\theta[/itex]/2)
Equating these two expressions: 2 sin([itex]\theta[/itex]/2) = [itex]\theta[/itex] or sin([itex]\theta[/itex]/2) = [itex]\theta[/itex]/2
I cannot find a way to simplify this expression further.
Any suggestions?
On a circle of radius 1, two points are marked: P1 and P2.
Two lines are drawn from the center of the circle:
one from the center to P1,
the other from the center to P2.
The angle between these two lines is [itex]\theta[/itex].
One more line is drawn: from P1 directly to P2. In other words, this third line is a chord on this circle.
For the special condition that the length of this chord equals the angle, find a simple expression.
i.e. – find a simple expression for [itex]\theta[/itex] given the special condition that chord length = [itex]\theta[/itex] = angle = [itex]\theta[/itex]
- - -
So far, all the expressions that I have worked out mix terms of [itex]\theta[/itex] and either sin([itex]\theta[/itex]) or cos([itex]\theta[/itex]); I have not been able to find an expression simply in terms of [itex]\theta[/itex], sin([itex]\theta[/itex]), or cos([itex]\theta[/itex]).
For example, following is one of my approaches:
Bisect the angle [itex]\theta[/itex], which also divides the chord in half.
The chord length is [itex]\theta[/itex].
But this value is also 2 sin([itex]\theta[/itex]/2)
Equating these two expressions: 2 sin([itex]\theta[/itex]/2) = [itex]\theta[/itex] or sin([itex]\theta[/itex]/2) = [itex]\theta[/itex]/2
I cannot find a way to simplify this expression further.
Any suggestions?