P
/|\
/ | \
/ | \ on this side there is an
/ | \ arc with A(θ) in the
/ | \ middle. the B(θ)
/θ | B(θ)\ triangle is not equal
O -------------- R to the θ triangle.
Q
the figure shows a setor of a circle with central angle theta. Let A(θ) be the area of the segment between the chord PR and the arc PR. Let B(θ) be the area of the triangle (right engle) PQR. Find lim and x->0+ A(θ)/B(θ)
1/3
A(theta)= (0.5*(r^2)*theta)-(r*sin theta)= 0.5(theta-sin theta)
B(theta)=(0.5*(r^2))*(0.5*r*cos theta* r*sin theta)=0.5sin theta(1- cos theta)
so the limit as theta approaches zero from the positive side of a(theta) over b (theta) is equal to:
[0.5(theta-sin theta)]/[0.5(sin theta (1- cos theta))]=
(theta - sin theta)/(sin theta - sin theta * cos theta)=
(theta - sin theta)/(sin theta - 0.5sin2 theta)=
Appling L'Hospitals rule:
the limit is:
(1- cos theta)/(cos theta - cos 2 theta)
using the rule again:
(sin theta)/(-sin theta + 2sin 2 theta)
and again:
(cos theta)/(-cos theta + 4 cos 2 theta)
=1/3
To find the limit of A(θ)/B(θ) as θ approaches 0 from the positive side, we can use the concept of limits to evaluate the ratio.
First, let's find the formulas for A(θ) and B(θ) in terms of θ.
The area A(θ) of the segment between the chord PR and the arc PR can be found using the formula:
A(θ) = (r^2 / 2) * (θ - sin(θ))
Here, r represents the radius of the sector.
The area B(θ) of the right triangle PQR can be found using the formula:
B(θ) = (1/2) * (PR) * (QR)
Since angle QPR is a right angle, we can use trigonometric ratios to find the lengths of PR and QR.
PR = r * cos(θ/2)
QR = r * sin(θ/2)
Substituting these values into the formula for B(θ), we get:
B(θ) = (1/2) * (r * cos(θ/2)) * (r * sin(θ/2))
Simplifying this expression:
B(θ) = (r^2 / 4) * sin(θ/2) * cos(θ/2)
Now, let's calculate the limit as x approaches 0 from the positive side:
lim (θ->0+) A(θ)/B(θ) = lim (θ->0+) [(r^2 / 2) * (θ - sin(θ))] / [(r^2 / 4) * sin(θ/2) * cos(θ/2)]
Canceling out the common terms:
lim (θ->0+) [(2θ - 2sin(θ)) / (sin(θ/2) * cos(θ/2))]
To evaluate this limit, we can use L'Hospital's rule. Differentiate the numerator and denominator with respect to θ:
lim (θ->0+) [2 - 2cos(θ)] / [(1/2) * (cos(θ/2) * cos(θ/2) - sin(θ/2) * sin(θ/2))]
Simplifying the expression:
lim (θ->0+) [2 - 2cos(θ)] / [(1/2) * (cos^2(θ/2) - sin^2(θ/2))]
Using the trigonometric identity cos^2(θ/2) - sin^2(θ/2) = cos(θ), we can further simplify:
lim (θ->0+) [2 - 2cos(θ)] / [(1/2) * cos(θ)]
Canceling out common terms:
lim (θ->0+) [4 - 4cos(θ)] / cos(θ)
Now we can substitute θ = 0 into the expression:
lim (θ->0+) [4 - 4cos(0)] / cos(0)
cos(0) = 1, so the expression becomes:
lim (θ->0+) [4 - 4] / 1
Simplifying further:
lim (θ->0+) 0 / 1 = 0
Therefore, the limit as θ approaches 0 from the positive side of A(θ)/B(θ) is equal to 0.
To find the limit as x approaches 0+ of A(θ)/B(θ), we first need to express A(θ) and B(θ) in terms of θ.
Let's start with A(θ), which represents the area of the segment between the chord PR and the arc PR. The formula for the area of a segment of a circle is:
A(θ) = (θ/360) * π * r^2 - 0.5 * r^2 * sin(θ)
Here, r represents the radius of the circle.
Now, let's determine B(θ), the area of the triangle PQR. Since PQ is a chord, it divides the angle θ into two equal angles. Let's call each of these angles φ. The formula for the area of a triangle given two sides and the included angle is:
B(θ) = 0.5 * PR * PQ * sin(φ)
Since PR is a radius of the circle (r) and PQ is 2 * r * sin(φ), we can simplify this expression:
B(θ) = 0.5 * r * (2 * r * sin(φ)) * sin(φ)
= r^2 * sin^2(φ)
Now that we have expressions for A(θ) and B(θ) in terms of θ and r, we can find the limit as θ approaches 0+ of A(θ)/B(θ):
lim(θ -> 0+) A(θ)/B(θ) = lim(θ -> 0+) [(θ/360) * π * r^2 - 0.5 * r^2 * sin(θ)] / [r^2 * sin^2(φ)]
Since θ is approaching 0, we can ignore the terms involving θ. Also, since φ is equal to θ/2, as θ approaches 0, φ also approaches 0.
lim(θ -> 0+) A(θ)/B(θ) = lim(θ -> 0+) [0 - 0] / [r^2 * sin^2(0)]
= 0 / (r^2 * 0)
= 0
Therefore, the limit as x approaches 0+ of A(θ)/B(θ) is 0.