Correct answer: A. R * (sin(theta) / (1 + sin(theta)))
Let the sector be defined by the vertex O and the arc connecting points A and B on the larger circle, such that the angle AOB is 2θ. The radius of this sector is OA = OB = R.
The inscribed circle has center C and radius r. Due to the symmetry of the figure, the center C must lie on the angle bisector of angle AOB. Let's call this line L. The line L divides the angle 2θ into two angles of size θ.
Let T be the point where the inscribed circle is tangent to one of the radial lines, say OA. The radius of the inscribed circle CT is perpendicular to the radius OA. This forms a right triangle, triangle OTC, with a right angle at T.
In the right triangle OTC:
- The angle at vertex O is θ.
- The side opposite this angle is CT, which is the radius r of the inscribed circle.
- The hypotenuse is OC, the distance from the sector's vertex to the center of the inscribed circle.
The sine of angle θ is the ratio of the opposite side to the hypotenuse:
sin(θ) = CT / OC = r / OC
From this, we can express the distance OC in terms of r and θ:
OC = r / sin(θ)
Now consider the line of symmetry L again. It passes through O, C, and the point where the inscribed circle is tangent to the arc of the sector. Let's call this point of tangency P. The points O, C, and P are collinear.
The distance OP is the radius of the large circle, R.
This distance can also be expressed as the sum of the distance OC and the distance CP.
The distance CP is the radius of the inscribed circle, r.
So, R = OP = OC + CP.
Substitute the expressions for OC and CP into this equation:
R = (r / sin(θ)) + r
Now, we solve this equation for r:
R = r · (1/sin(θ) + 1)
Factor out r.
R = r · ( (1 + sin(θ)) / sin(θ) )
Find a common denominator for the terms in the parentheses.
To isolate r, multiply R by the reciprocal of the expression in the parentheses:
r = R · ( sin(θ) / (1 + sin(θ)) )
Therefore, the correct choice is A.
Choice B is incorrect. This expression would result from an incorrect geometric setup, specifically by assuming R = OC - CP, which implies the point of tangency P is between O and C rather than C being between O and P.
Choice C is incorrect. This expression would result from using cosine instead of sine in the trigonometric ratio for the right triangle OTC, a common error in trig applications.
Choice D is incorrect. This results from the oversimplified assumption that OC = R, which would mean the center of the inscribed circle lies on the arc of the larger circle. This ignores the fact that the inscribed circle's own radius r must be accounted for.