1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

2 Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \begin{figure}[h] \end{figure}

1. (A) Calculate AC and angle ACB . Hence calculate AD .
   (B) Calculate the area of the garden.
2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M , of side FG , and sector angle 1.1 radians, as shown. \(\mathrm { FG } = 1.8 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-2_579_981_1512_567} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure} Calculate the area of one of these fence panels.

5 Solve the equation \(\sin 2 \theta = 0.7\) for values of \(\theta\) between 0 and \(2 \pi\), giving your answers in radians correct to 3 significant figures.

1 In Fig. 6, OAB is a thin bent rod, with \(\mathrm { OA } = a\) metres, \(\mathrm { AB } = b\) metres and angle \(\mathrm { OAB } = 120 ^ { \circ }\). The bent rod lies in a vertical plane. OA makes an angle \(\theta\) above the horizontal. The vertical height BD of B above O is \(h\) metres. The horizontal through A meets BD at C and the vertical through A meets OD at E . \begin{figure}[h] \end{figure}

1. Find angle BAC in terms of \(\theta\). Hence show that $$h = a \sin \theta + b \sin \left( \theta - 60 ^ { \circ } \right) .$$
2. Hence show that \(h = \left( a + \frac { 1 } { 2 } b \right) \sin \theta - \frac { \sqrt { 3 } } { 2 } b \cos \theta\). The rod now rotates about O , so that \(\theta\) varies. You may assume that the formulae for \(h\) in parts (i) and (ii) remain valid.
3. Show that OB is horizontal when \(\tan \theta = \frac { \sqrt { 3 } b } { 2 a + b }\). In the case when \(a = 1\) and \(b = 2 , h = 2 \sin \theta - \sqrt { 3 } \cos \theta\).
4. Express \(2 \sin \theta - \sqrt { 3 } \cos \theta\) in the form \(R \sin ( \theta - \alpha )\). Hence, for this case, write down the maximum value of \(h\) and the corresponding value of \(\theta\).

7. \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle \(C _ { 1 }\) has centre \(O\) and radius \(R\). The tangents \(A P\) and \(B P\) to \(C _ { 1 }\) meet at the point \(P\) and angle \(A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }\). A sequence of circles \(C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots\) is drawn so that each new circle \(C _ { n + 1 }\) touches each of \(C _ { n } , A P\) and \(B P\) for \(n = 1,2,3 , \ldots\) as shown in Figure 2. The centre of each circle lies on the line \(O P\).

1. Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
2. Find, in terms of \(R\) and \(\alpha\), the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral \(P A O B\), not enclosed by part of \(C _ { 1 }\) or any of the other circles, is \(S\).
3. Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
4. Show that, as \(\alpha\) varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
5. Find, in terms of \(R\), the least value of \(S\) for \(\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }\).

2 \includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-2_311_521_651_810} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 7 cm . The angle \(A O B\) is \(140 ^ { \circ }\).

1. Express \(140 ^ { \circ }\) in radians, giving your answer in an exact form as simply as possible.
2. Find the perimeter of the segment shaded in the diagram, giving your answer correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_469_673_1720_737} The diagram shows triangle \(A B C\), with \(A B = 10 \mathrm {~cm} , B C = 13 \mathrm {~cm}\) and \(C A = 14 \mathrm {~cm} . E\) and \(F\) are points on \(A B\) and \(A C\) respectively such that \(A E = A F = 4 \mathrm {~cm}\). The sector \(A E F\) of a circle with centre \(A\) is removed to leave the shaded region \(E B C F\).

1. Show that angle \(C A B\) is 1.10 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region \(E B C F\).
3. Find the area of the shaded region \(E B C F\).

8 \includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-03_420_729_1027_708} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 5 cm . Angle \(A O B\) is \(\theta\) radians. The area of triangle \(A O B\) is \(8 \mathrm {~cm} ^ { 2 }\).

1. Given that the angle \(\theta\) is obtuse, find \(\theta\). The shaded segment in the diagram is bounded by the chord \(A B\) and the arc \(A B\).
2. Find the area of the segment, giving your answer correct to 3 significant figures.
3. Find the perimeter of the segment, giving your answer correct to 3 significant figures.

1 \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_319_454_246_810} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 12 cm . The reflex angle \(A O B\) is 4.2 radians.

1. Find the perimeter of the sector.
2. Find the area of the sector.

7 \includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-3_412_707_751_680} The diagram shows two circles of radius 7 cm with centres \(A\) and \(B\). The distance \(A B\) is 12 cm and the point \(C\) lies on both circles. The region common to both circles is shaded.

1. Show that angle \(C A B\) is 0.5411 radians, correct to 4 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

8 \begin{figure}[h] \end{figure} Fig. 1 shows a sector \(A O B\) of a circle, centre \(O\) and radius \(O A\). The angle \(A O B\) is 1.2 radians and the area of the sector is \(60 \mathrm {~cm} ^ { 2 }\).

1. Find the perimeter of the sector. A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector \(A O B\) from Fig. 1, and the area of each successive sector is \(\frac { 3 } { 5 }\) of the area of the previous one. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. (a) Find the area of the fifth sector in the pattern.
   (b) Find the total area of the first ten sectors in the pattern.
   (c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.

5 \includegraphics[max width=\textwidth, alt={}, center]{570435e0-5685-4c5b-9ed8-f2bc22bdfb24-02_396_1070_1768_536} The diagram shows two congruent triangles, \(B C D\) and \(B A E\), where \(A B C\) is a straight line. In triangle \(B C D , B D = 8 \mathrm {~cm} , C D = 11 \mathrm {~cm}\) and angle \(C B D = 65 ^ { \circ }\). The points \(E\) and \(D\) are joined by an arc of a circle with centre \(B\) and radius 8 cm .

1. Find angle \(B C D\).
2. (a) Show that angle \(E B D\) is 0.873 radians, correct to 3 significant figures.
   (b) Hence find the area of the shaded segment bounded by the chord \(E D\) and the arc \(E D\), giving your answer correct to 3 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-2_515_501_1439_822} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius 8 cm . The perimeter of the sector is 23.2 cm .

1. Find angle \(A O B\) in radians.
2. Find the area of the sector.

3 \includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-2_436_526_762_767} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B\) is \(72 ^ { \circ }\).

1. Express \(72 ^ { \circ }\) exactly in radians, simplifying your answer. The area of the sector \(A O B\) is \(45 \pi \mathrm {~cm} ^ { 2 }\).
2. Find the value of \(r\).
3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\), giving your answer correct to 3 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-2_405_688_1535_685} The diagram shows a sector \(B A C\) of a circle with centre \(A\) and radius 16 cm . The angle \(B A C\) is 0.8 radians. The length \(A D\) is 7 cm .

1. Find the area of the region \(B D C\).
2. Find the perimeter of the region \(B D C\).

3 \includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_350_597_1695_735} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 12 cm . The angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians.

1. Find the exact length of the \(\operatorname { arc } A B\).
2. Find the exact area of the shaded segment enclosed by the arc \(A B\) and the chord \(A B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-2_576_599_1062_733} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 8 cm . The angle \(A O B\) is 1.2 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively such that \(O C = 5.2 \mathrm {~cm}\) and \(O D = 2.6 \mathrm {~cm} . C D\) is a straight line.

1. Find the area of the shaded region \(A C D B\).
2. Find the perimeter of the shaded region \(A C D B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_417_476_1030_790} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is \(54 ^ { \circ }\). The perimeter of the sector is 60 cm .

1. Express \(54 ^ { \circ }\) exactly in radians, simplifying your answer.
2. Find the value of \(r\), giving your answer correct to 3 significant figures.

11

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_469_878_274_671} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} Fig. 11.1 shows the surface ABCD of a TV presenter's desk. AB and CD are arcs of circles with centre O and sector angle 2.5 radians. \(\mathrm { OC } = 60 \mathrm {~cm}\) and \(\mathrm { OB } = 140 \mathrm {~cm}\).
   (A) Calculate the length of the arc CD.
   (B) Calculate the area of the surface ABCD of the desk.
2. The TV presenter is at point P , shown in Fig. 11.2. A TV camera can move along the track EF , which is of length 3.5 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_378_877_1334_675} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} When the camera is at E , the TV presenter is 1.6 m away. When the camera is at F , the TV presenter is 2.8 m away.
   (A) Calculate, in degrees, the size of angle EFP.
   (B) Calculate the shortest possible distance between the camera and the TV presenter.

9 Charles has a slice of cake; its cross-section is a sector of a circle, as shown in Fig. 9. The radius is \(r \mathrm {~cm}\) and the sector angle is \(\frac { \pi } { 6 }\) radians. He wants to give half of the slice to Jan. He makes a cut across the sector as shown. \begin{figure}[h] \end{figure} Show that when they each have half the slice, \(a = r \sqrt { \frac { \pi } { 6 } }\). Section B (36 marks)

13 \begin{figure}[h] \end{figure} In a concert hall, seats are arranged along arcs of concentric circles, as shown in Fig. 13.1. As shown in Fig. 13.2, the stage is part of a sector ABO of radius 11 m . Fig. 13.2 also gives the dimensions of the stage.

1. Show that angle \(\mathrm { COD } = 1.55\) radians, correct to 2 decimal places. Hence find the area of the stage.
2. There are four rows of seats, with their backs along arcs, with centre O, of radii \(7.4 \mathrm {~m} , 8.6 \mathrm {~m} , 9.8 \mathrm {~m}\) and 11 m . Each seat takes up 80 cm of the arc.
   (A) Calculate how many seats can fit in the front row.
   (B) Calculate how many more seats can fit in the back row than the front row.

13 Fig. 13.1 shows a greenhouse which is built against a wall. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The greenhouse is a prism of length 5.5 m . The curve AC is an arc of a circle with centre B and radius 2.1 m , as shown in Fig. 13.2. The sector angle ABC is 1.8 radians and ABD is a straight line. The curved surface of the greenhouse is covered in polythene.

1. Find the length of the arc AC and hence find the area of polythene required for the curved surface of the greenhouse.
2. Calculate the length BD .
3. Calculate the volume of the greenhouse.

5 A sector of a circle has angle 1.6 radians and area \(45 \mathrm {~cm} ^ { 2 }\). Find the radius and perimeter of the sector.

4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.

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 \(\theta _ { 1 } = 0.5\).