Derive equation from area/geometry

4 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-2_358_618_1082_762} The diagram shows a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The chord \(A B\) divides the sector into a triangle \(A O B\) and a segment \(A X B\). Angle \(A O B\) is \(\theta\) radians.

1. In the case where the areas of the triangle \(A O B\) and the segment \(A X B\) are equal, find the value of the constant \(p\) for which \(\theta = p \sin \theta\).
2. In the case where \(r = 8\) and \(\theta = 2.4\), find the perimeter of the segment \(A X B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-2_355_601_1562_772} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of triangle \(A O B\) is half the area of the sector.

1. Show that \(\alpha\) satisfies the equation $$x = 2 \sin x$$
2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \sin x _ { n } \right)$$ converges, then it converges to a root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 1.8\), to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

3 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-2_337_828_657_657} In the diagram, \(A B C D\) is a rectangle with \(A B = 3 a\) and \(A D = a\). A circular arc, with centre \(A\) and radius \(r\), joins points \(M\) and \(N\) on \(A B\) and \(C D\) respectively. The angle \(M A N\) is \(x\) radians. The perimeter of the sector \(A M N\) is equal to half the perimeter of the rectangle.

1. Show that \(x\) satisfies the equation $$\sin x = \frac { 1 } { 4 } ( 2 + x ) \text {. }$$
2. This equation has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 + x _ { n } } { 4 } \right) ,$$ with initial value \(x _ { 1 } = 0.8\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-3_380_641_258_751} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The angle \(B O C\) is \(x\) radians. The area of the shaded segment is a quarter of the area of the semicircle.

1. Show that \(x\) satisfies the equation $$x = \frac { 3 } { 4 } \pi - \sin x$$
2. This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
3. Use the iterative formula $$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{cc85b13a-7f15-4025-a545-373cda454de8-3_456_495_255_824} The diagram shows a circle with centre \(O\) and radius 10 cm . The chord \(A B\) divides the circle into two regions whose areas are in the ratio \(1 : 4\) and it is required to find the length of \(A B\). The angle \(A O B\) is \(\theta\) radians.

1. Show that \(\theta = \frac { 2 } { 5 } \pi + \sin \theta\).
2. Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1 , to find \(\theta\) correct to 2 decimal places. Hence find the length of \(A B\) in centimetres correct to 1 decimal place.

4 \includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-2_339_1395_1089_374} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The tangent at \(C\) meets \(A B\) produced at \(T\). The angle \(B O C\) is \(x\) radians. The area of the shaded region is equal to the area of the semicircle.

1. Show that \(x\) satisfies the equation $$\tan x = x + \pi$$
2. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } + \pi \right)\) to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689} In the diagram, \(A B C\) is a triangle in which angle \(A B C\) is a right angle and \(B C = a\). A circular arc, with centre \(C\) and radius \(a\), joins \(B\) and the point \(M\) on \(A C\). The angle \(A C B\) is \(\theta\) radians. The area of the sector \(C M B\) is equal to one third of the area of the triangle \(A B C\).

1. Show that \(\theta\) satisfies the equation $$\tan \theta = 3 \theta .$$
2. This equation has one root in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). Use the iterative formula $$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{d1377d66-73c8-4d97-9cae-d784b41fb0a8-2_519_800_1359_669} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and the angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The perimeter of the shaded region is equal to the circumference of the circle.

1. Show that \(x\) satisfies the equation $$\tan x = \pi - x .$$
2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.3.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi - x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-08_492_812_260_664} In the diagram, \(A\) is the mid-point of the semicircle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the semicircle at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The area of the shaded region bounded by \(A B , A C\) and the arc with centre \(A\) is equal to half the area of the semicircle.

1. Use triangle \(O A B\) to show that \(A B = 2 r \cos x\).
2. Hence show that \(x = \cos ^ { - 1 } \sqrt { } \left( \frac { \pi } { 16 x } \right)\).
3. Verify by calculation that \(x\) lies between 1 and 1.5.
4. Use an iterative formula based on the equation in part (ii) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{b89c016e-dc56-48f4-b4c4-b432418e1b28-3_435_672_273_684} The diagram shows a curved rod \(A B\) of length 100 cm which forms an arc of a circle. The end points \(A\) and \(B\) of the rod are 99 cm apart. The circle has radius \(r \mathrm {~cm}\) and the arc \(A B\) subtends an angle of \(2 \alpha\) radians at \(O\), the centre of the circle.

1. Show that \(\alpha\) satisfies the equation \(\frac { 99 } { 100 } x = \sin x\).
2. Given that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\), verify by calculation that this root lies between 0.1 and 0.5.
3. Show that if the sequence of values given by the iterative formula $$x _ { n + 1 } = 50 \sin x _ { n } - 48.5 x _ { n }$$ converges, then it converges to a root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.

5 \includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-2_396_392_1603_879} The diagram shows a chord joining two points, \(A\) and \(B\), on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of the shaded segment is one sixth of the area of the circle.

1. Show that \(\alpha\) satisfies the equation $$x = \frac { 1 } { 3 } \pi + \sin x .$$
2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \pi + \sin x _ { n } ,$$ with initial value \(x _ { 1 } = 2\), to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{da8dedae-4714-408e-a983-90ece63d9e91-08_501_1086_262_525} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The area of the shaded region is equal to the area of the circle.

1. Show that \(x\) satisfies the equation \(\tan x = \pi + x\).
2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.4.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi + x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-10_627_611_255_767} The diagram shows a circle with centre \(O\) and radius \(r\). The angle of the minor sector \(A O B\) of the circle is \(x\) radians. The area of the major sector of the circle is 3 times the area of the shaded region.

1. Show that \(x = \frac { 3 } { 4 } \sin x + \frac { 1 } { 2 } \pi\).
2. Show by calculation that the root of the equation in (a) lies between 2 and 2.5.
3. Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{3c63c42a-2658-4984-93e8-b2a8d18eb37a-14_407_734_267_699} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The shaded region is the minor segment on the chord \(A C\) and its area is one third of the area of the semicircle. The angle \(C A B\) is \(\theta\) radians.

1. Show that \(\theta = \frac { 1 } { 3 } ( \pi - 1.5 \sin 2 \theta )\).
2. Verify by calculation that \(0.5 < \theta < 0.7\).
3. Use an iterative formula based on the equation in part (a) to determine \(\theta\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

10 \includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-6_405_661_251_703} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 6 cm .
The angle \(A O B\) is \(\theta\) radians.
The area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\) is \(7.2 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.4 + \sin \theta\).
2. Let \(\mathrm { F } ( \theta ) = 0.4 + \sin \theta\). By considering the value of \(\mathrm { F } ^ { \prime } ( \theta )\) where \(\theta = 1.2\), explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that 1.2 is sufficiently close to the root.
3. Use the iterative formula \(\theta _ { n + 1 } = 0.4 + \sin \theta _ { n }\) with a starting value of 1.2 to find the value of \(\theta\) correct to 4 significant figures.
   You should show the result of each iteration.
4. Use a change of sign method to show that the value of \(\theta\) found in part (c) is correct to 4 significant figures.

10 \includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-7_545_659_255_244} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(O A\). The angle \(A O B\) is \(\theta\) radians. \(M\) is the mid-point of \(O A\). The ratio of areas \(O M B : M A B\) is 2:3.

1. Show that \(\theta = 1.25 \sin \theta\). The equation \(\theta = 1.25 \sin \theta\) has only one root for \(\theta > 0\).
2. This root can be found by using the iterative formula \(\theta _ { n + 1 } = 1.25 \sin \theta _ { n }\) with a starting value of \(\theta _ { 1 } = 0.5\).

8 The diagram shows a sector of a circle \(O A B\). \(C\) is the midpoint of \(O B\).
Angle \(A O B\) is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-10_700_963_536_534} 8

1. Given that the area of the triangle \(O A C\) is equal to one quarter of the area of the sector \(O A B\), show that \(\theta = 2 \sin \theta\) 8
2. Use the Newton-Raphson method with \(\theta _ { 1 } = \pi\), to find \(\theta _ { 3 }\) as an approximation for \(\theta\). Give your answer correct to five decimal places.
   8
3. Given that \(\theta = 1.89549\) to five decimal places, find an estimate for the percentage error in the approximation found in part (b).
   Turn over for the next question

6 \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-3_545_557_269_794} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) lies on \(O B\), and \(A C\) is perpendicular to \(O B\). The area of the triangle \(A O C\) is equal to the area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\).

1. Show that \(\theta = \sin \theta ( 1 + \cos \theta )\). The equation \(\theta = \sin \theta ( 1 + \cos \theta )\) has only one positive root.
2. Use an iterative process based on this equation to find the value of the root correct to 3 significant figures. Use a starting value of 1 and show the result of each iteration. Use a change of sign to verify that the value you have found is correct to 3 significant figures.

\includegraphics{figure_6} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(\angle OAB\) is equal to \(x\) radians. The shaded region is bounded by \(AB\), \(AC\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.

1. Show that \(x = \cos^{-1}\left(\frac{\pi}{4 + 4x}\right)\). [3]
2. Verify by calculation that \(x\) lies between 1 and 1.5. [2]
3. Use the iterative formula $$x_{n+1} = \cos^{-1}\left(\frac{\pi}{4 + 4x_n}\right)$$ to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

\includegraphics{figure_6} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(OAB\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.

1. Show that \(\cos 2\theta = \frac{2\sin 2\theta - \pi}{4\theta}\). [5]
2. Use the iterative formula $$\theta_{n+1} = \frac{1}{2}\cos^{-1}\left(\frac{2\sin 2\theta_n - \pi}{4\theta_n}\right),$$ with initial value \(\theta_1 = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3]