1.05b Sine and cosine rules: including ambiguous case

5 \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-06_761_460_258_840} The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre \(O\) and radius 5 cm . The thickness of the cord and the size of the pin \(P\) can be neglected. The pin is situated 13 cm vertically below \(O\). Points \(A\) and \(B\) are on the circumference of the circle such that \(A P\) and \(B P\) are tangents to the circle. The cord passes over the major arc \(A B\) of the circle and under the pin such that the cord is taut. Calculate the length of the cord.

8 \includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-10_454_744_255_703} The diagram shows triangle \(A B C\) in which angle \(B\) is a right angle. The length of \(A B\) is 8 cm and the length of \(B C\) is 4 cm . The point \(D\) on \(A B\) is such that \(A D = 5 \mathrm {~cm}\). The sector \(D A C\) is part of a circle with centre \(D\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-10_348_700_262_721} In the diagram, \(A B C\) is an isosceles triangle with \(A B = B C = r \mathrm {~cm}\) and angle \(B A C = \theta\) radians. The point \(D\) lies on \(A C\) and \(A B D\) is a sector of a circle with centre \(A\).

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 10\) and \(\theta = 0.6\), find the perimeter of the shaded region.

9 \includegraphics[max width=\textwidth, alt={}, center]{3b44e558-f91d-4175-acda-eceb70dad82c-12_497_652_260_744} In the diagram, arc \(A B\) is part of a circle with centre \(O\) and radius 8 cm . Arc \(B C\) is part of a circle with centre \(A\) and radius 12 cm , where \(A O C\) is a straight line.

1. Find angle \(B A O\) in radians.
2. Find the area of the shaded region.
3. Find the perimeter of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-08_556_751_255_696} In the diagram the lengths of \(A B\) and \(A C\) are both 15 cm . The point \(P\) is the foot of the perpendicular from \(C\) to \(A B\). The length \(C P = 9 \mathrm {~cm}\). An arc of a circle with centre \(B\) passes through \(C\) and meets \(A B\) at \(Q\).

1. Show that angle \(A B C = 1.25\) radians, correct to 3 significant figures.
2. Calculate the area of the shaded region which is bounded by the \(\operatorname { arc } C Q\) and the lines \(C P\) and \(P Q\).

5 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733} In the diagram, \(X\) and \(Y\) are points on the line \(A B\) such that \(B X = 9 \mathrm {~cm}\) and \(A Y = 11 \mathrm {~cm}\). Arc \(B C\) is part of a circle with centre \(X\) and radius 9 cm , where \(C X\) is perpendicular to \(A B\). Arc \(A C\) is part of a circle with centre \(Y\) and radius 11 cm .

1. Show that angle \(X Y C = 0.9582\) radians, correct to 4 significant figures.
2. Find the perimeter of \(A B C\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813} The diagram shows the circular cross-section of a uniform cylindrical log with centre \(O\) and radius 20 cm . The points \(A , X\) and \(B\) lie on the circumference of the cross-section and \(A B = 32 \mathrm {~cm}\).

1. Show that angle \(A O B = 1.855\) radians, correct to 3 decimal places.
2. Find the area of the sector \(A X B O\). The section \(A X B C D\), where \(A B C D\) is a rectangle with \(A D = 18 \mathrm {~cm}\), is removed.
3. Find the area of the new cross-section (shown shaded in the diagram).

6 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-2_389_995_1432_575} In the diagram, \(A B C\) is a triangle in which \(A B = 4 \mathrm {~cm} , B C = 6 \mathrm {~cm}\) and angle \(A B C = 150 ^ { \circ }\). The line \(C X\) is perpendicular to the line \(A B X\).

1. Find the exact length of \(B X\) and show that angle \(C A B = \tan ^ { - 1 } \left( \frac { 3 } { 4 + 3 \sqrt { } 3 } \right)\).
2. Show that the exact length of \(A C\) is \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).

1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-2_663_446_1562_847} In the diagram, triangle \(A B C\) is right-angled at \(C\) and \(M\) is the mid-point of \(B C\). It is given that angle \(A B C = \frac { 1 } { 3 } \pi\) radians and angle \(B A M = \theta\) radians. Denoting the lengths of \(B M\) and \(M C\) by \(x\),

1. find \(A M\) in terms of \(x\),
2. show that \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-3_655_1011_255_566} The diagram shows triangle \(A B C\) where \(A B = 5 \mathrm {~cm} , A C = 4 \mathrm {~cm}\) and \(B C = 3 \mathrm {~cm}\). Three circles with centres at \(A , B\) and \(C\) have radii \(3 \mathrm {~cm} , 2 \mathrm {~cm}\) and 1 cm respectively. The circles touch each other at points \(E , F\) and \(G\), lying on \(A B , A C\) and \(B C\) respectively. Find the area of the shaded region \(E F G\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-3_602_570_260_790} In the diagram, triangle \(A B C\) is right-angled and \(D\) is the mid-point of \(B C\). Angle \(D A C = 30 ^ { \circ }\) and angle \(B A D = x ^ { \circ }\). Denoting the length of \(A D\) by \(l\),

1. express each of \(A C\) and \(B C\) exactly in terms of \(l\), and show that \(A B = \frac { 1 } { 2 } l \sqrt { } 7\),
2. show that \(x = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { } 3 } \right) - 30\).

3 \includegraphics[max width=\textwidth, alt={}, center]{933cdfe1-27bb-450d-8b9a-b494916242cb-2_737_693_1484_726} In the diagram, \(A B E D\) is a trapezium with right angles at \(E\) and \(D\), and \(C E D\) is a straight line. The lengths of \(A B\) and \(B C\) are \(2 d\) and \(( 2 \sqrt { 3 } ) d\) respectively, and angles \(B A D\) and \(C B E\) are \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively.

1. Find the length of \(C D\) in terms of \(d\).
2. Show that angle \(C A D = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { 3 } } \right)\).

5 \includegraphics[max width=\textwidth, alt={}, center]{9cdb00a6-1e86-4185-bb73-ed3ecab981ba-3_560_506_258_822} The diagram shows a metal plate \(O A B C\), consisting of a right-angled triangle \(O A B\) and a sector \(O B C\) of a circle with centre \(O\). Angle \(A O B = 0.6\) radians, \(O A = 6 \mathrm {~cm}\) and \(O A\) is perpendicular to \(O C\).

1. Show that the length of \(O B\) is 7.270 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

5 \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-08_446_844_260_648} The diagram shows an isosceles triangle \(A B C\) in which \(A C = 16 \mathrm {~cm}\) and \(A B = B C = 10 \mathrm {~cm}\). The circular arcs \(B E\) and \(B D\) have centres at \(A\) and \(C\) respectively, where \(D\) and \(E\) lie on \(A C\).

1. Show that angle \(B A C = 0.6435\) radians, correct to 4 decimal places.
2. Find the area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_462_721_1672_712} Two forces have magnitudes \(P \mathrm {~N}\) and \(Q \mathrm {~N}\). The resultant of the two forces has magnitude 12 N and acts in a direction \(40 ^ { \circ }\) clockwise from the force of magnitude \(P \mathrm {~N}\) and \(80 ^ { \circ }\) anticlockwise from the force of magnitude \(Q \mathrm {~N}\) (see diagram). Find the value of \(Q\).

7 A particle \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = k x - \frac { \left( 1 + k ^ { 2 } \right) x ^ { 2 } } { 245 }$$ where \(k\) is a constant. \(P\) passes through the points \(A ( 14 , a )\) and \(B ( 42,2 a )\), where \(a\) is a constant.

1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is \(63.435 ^ { \circ }\), correct to 3 decimal places. For the larger angle of projection, calculate
2. the time after projection when \(P\) passes through \(A\),
3. the speed and direction of motion of \(P\) when it passes through \(B\). {www.cie.org.uk} after the live examination series. }

7. \begin{figure}[h] \end{figure} Not to scale Figure 3 shows the design for a structure used to support a roof. The structure consists of four wooden beams, \(A B , B D , B C\) and \(A D\). Given \(A B = 6.5 \mathrm {~m} , B C = B D = 4.7 \mathrm {~m}\) and angle \(B A C = 35 ^ { \circ }\)

1. find, to one decimal place, the size of angle \(A C B\),
2. find, to the nearest metre, the total length of wood required to make this structure.

5. \begin{figure}[h] \end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\). Given that

• \(O A = O C = 6 \mathrm {~m}\)
• \(A D = 14 \mathrm {~m}\)
• angle \(A D C = 0.43\) radians
• angle \(A O D\) is an obtuse angle
• \(O C D\) is a straight line
  find
  1. the size of angle \(A O D\), in radians, to 3 decimal places,
  2. the length of arc \(A B C\), in metres, to one decimal place,
  3. the total area of the viewing platform, in \(\mathrm { m } ^ { 2 }\), to one decimal place.

1. The triangle \(A B C\) is such that

• \(A B = 15 \mathrm {~cm}\)
• \(A C = 25 \mathrm {~cm}\)
• angle \(B A C = \theta ^ { \circ }\)
• area triangle \(A B C = 100 \mathrm {~cm} ^ { 2 }\)
  1. Find the value of \(\sin \theta ^ { \circ }\)

Given that \(\theta > 90\)

find the length of \(B C\), in cm , to 3 significant figures.

7. \begin{figure}[h] \end{figure} The shape \(A B C D A\) consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\), as shown in Figure 2. The point \(D\) lies on \(O C\).
The radius of the circle is 6 cm , length \(A D\) is 5 cm and angle \(A O D\) is 0.7 radians.

1. Find the area of the sector \(A B C O A\), giving your answer to one decimal place. Given angle \(A D O\) is obtuse,
2. find the size of angle \(A D O\), giving your answer to 3 decimal places.
3. Hence find the perimeter of shape \(A B C D A\), giving your answer to one decimal place.
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1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle \(A B C\) with

• \(A B = p\) metres
• \(A C = q\) metres
• \(B C = 2 \sqrt { 2 }\) metres
• angle \(B A C = 60 ^ { \circ }\)
  1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side \(A C\) is 2 metres longer than side \(A B\), use algebra to find

1. the exact value of \(p\),
2. the exact value of \(q\). Using the answers to part (b),

calculate the exact area of the flower bed.

2. In the triangle \(A B C\),

• \(A B = 21 \mathrm {~cm}\)
• \(B C = 13 \mathrm {~cm}\)
• angle \(B A C = 25 ^ { \circ }\)
• angle \(A C B = x ^ { \circ }\)
  1. Use the sine rule to find the value of \(\sin x ^ { \circ }\), giving your answer to 4 decimal places.

Given also that \(A B\) is the longest side of the triangle,

find the value of \(x\), giving your answer to 2 decimal places.

5. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\) Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)

1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

4. A parallelogram \(A B C D\) has area \(40 \mathrm {~cm} ^ { 2 }\) Given that \(A B\) has length \(10 \mathrm {~cm} , B C\) has length 6 cm and angle \(D A B\) is obtuse, find

1. the size of angle \(D A B\), in degrees, to 2 decimal places,
2. the length of diagonal \(B D\), in cm , to one decimal place.