1.05b Sine and cosine rules: including ambiguous case

4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.

2 The diagram shows a triangle \(A B C\) and the arc \(A B\) of a circle whose centre is \(C\) and whose radius is 24 cm . \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-2_506_403_1187_781} The length of the side \(A B\) of the triangle is 32 cm . The size of the angle \(A C B\) is \(\theta\) radians.

1. Show that \(\theta = 1.46\) correct to three significant figures.
2. Calculate the length of the \(\operatorname { arc } A B\) to the nearest cm .
3. 1. Calculate the area of the sector \(A B C\) to the nearest \(\mathrm { cm } ^ { 2 }\).
   2. Hence calculate the area of the shaded segment to the nearest \(\mathrm { cm } ^ { 2 }\).

4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.

3 The diagram shows a triangle \(A B C\). The length of \(A C\) is 18.7 cm , and the sizes of angles \(B A C\) and \(A B C\) are \(72 ^ { \circ }\) and \(50 ^ { \circ }\) respectively.

1. Show that the length of \(B C = 23.2 \mathrm {~cm}\), correct to the nearest 0.1 cm .
2. Calculate the area of triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).

3 The diagram shows a triangle \(A B C\). The size of angle \(A\) is \(63 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.4 m and 5.26 m respectively.

1. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
2. Show that the length of \(B C\) is 6.86 m , correct to three significant figures.
3. Find the value of \(\sin \boldsymbol { B }\) to two significant figures.

3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 5 \mathrm {~cm} , A C = 8 \mathrm {~cm}\), \(B C = 10 \mathrm {~cm}\) and angle \(B A C = \theta\).

1. Show that \(\theta = 97.9 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. 1. Calculate the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
   2. The line through \(A\), perpendicular to \(B C\), meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer, in cm , to three significant figures.

4 The triangle \(A B C\), shown in the diagram, is such that \(A B\) is 10 metres and angle \(B A C\) is \(150 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-3_323_746_406_648} The area of triangle \(A B C\) is \(40 \mathrm {~m} ^ { 2 }\).

1. Show that the length of \(A C\) is 16 metres.
2. Calculate the length of \(B C\), giving your answer, in metres, to two decimal places.
3. Calculate the smallest angle of triangle \(A B C\), giving your answer to the nearest \(0.1 ^ { \circ }\).

3 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-3_273_622_356_708} The lengths of \(A C\) and \(B C\) are 5 cm and 6 cm respectively.
The area of triangle \(A B C\) is \(12.5 \mathrm {~cm} ^ { 2 }\), and angle \(A C B\) is obtuse.

1. Find the size of angle \(A C B\), giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Find the length of \(A B\), giving your answer to two significant figures.

1 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_423_707_612_657} The lengths of \(A C\) and \(B C\) are 5 cm and 4.8 cm respectively.
The size of the angle \(B C A\) is \(30 ^ { \circ }\).

1. Calculate the area of the triangle \(A B C\).
2. Calculate the length of \(A B\), giving your answer to three significant figures.

2 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_757_558_1409_726} The lengths of \(A C\) and \(B C\) are 4.8 cm and 12 cm respectively.
The size of the angle \(B A C\) is \(100 ^ { \circ }\).

1. Show that angle \(A B C = 23.2 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to three significant figures.

2 The diagram shows a shaded segment of a circle with centre \(O\) and radius 14 cm , where \(P Q\) is a chord of the circle. \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-2_423_551_1270_740} In triangle \(O P Q\), angle \(P O Q = \frac { 3 \pi } { 7 }\) radians and angle \(O P Q = \alpha\) radians.

1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
2. Find \(\alpha\) in terms of \(\pi\).
3. Find the perimeter of the shaded segment, giving your answer to three significant figures.

4 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-3_394_522_1062_751} The size of angle \(B A C\) is \(65 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.6 m and 8.3 m respectively.

1. Show that the length of \(B C\) is 8.56 m , correct to three significant figures.
2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
3. The perpendicular from \(A\) to \(B C\) meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer to the nearest 0.1 m .

3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 6 \mathrm {~cm} , B C = 15 \mathrm {~cm}\), angle \(B A C = 150 ^ { \circ }\) and angle \(A C B = \theta\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-4_376_867_406_584}

1. Show that \(\theta = 11.5 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to three significant figures.

1 The triangle \(A B C\), shown in the diagram, is such that \(A C = 9 \mathrm {~cm} , B C = 10 \mathrm {~cm}\), angle \(A B C = 54 ^ { \circ }\) and the acute angle \(B A C = \theta\).

1. Show that \(\theta = 64 ^ { \circ }\), correct to the nearest degree.
2. Calculate the area of triangle \(A B C\), giving your answer to the nearest square centimetre.

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_341_371_968_815} The radius of the circle is 20 cm and the angle \(A O B = 0.8\) radians.

1. Find the length of the arc \(A B\).
2. Find the area of the sector \(O A B\).
3. A line from \(B\) meets the radius \(O A\) at the point \(D\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_344_371_1747_815} The length of \(B D\) is 15 cm . Find the size of the obtuse angle \(O D B\), in radians, giving your answer to three significant figures.

1 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(47 ^ { \circ }\) and the lengths of \(A B\) and \(A C\) are 5 cm and 12 cm respectively.

1. Calculate the area of the triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).
2. Calculate the length of \(B C\), giving your answer, in cm , to one decimal place.
   [0pt] [3 marks]

2 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(72 ^ { \circ }\) and the size of angle \(A B C\) is \(48 ^ { \circ }\). The length of \(B C\) is 20 cm .

1. Show that the length of \(A C\) is 15.6 cm , correct to three significant figures.
2. The midpoint of \(B C\) is \(M\). Calculate the length of \(A M\), giving your answer, in cm , to three significant figures.
   [0pt] [4 marks]

6 The diagram shows a triangle \(A B C\). The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.

1. Show that \(\theta = 0.586\), correct to three significant figures.
2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-14_467_677_1462_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
   [0pt] [6 marks]

8. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm} , C D = 8 \mathrm {~cm}\), \(A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).

1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
2. Find the size of \(\angle B C D\) in degrees.
3. Find the area of quadrilateral \(A B C D\).

6. \begin{figure}[h] \end{figure} Figure 1 shows triangle \(A B C\) in which \(A C = 8 \mathrm {~cm}\) and \(\angle B A C = \angle B C A = 30 ^ { \circ }\).

1. Find the area of triangle \(A B C\) in the form \(k \sqrt { 3 }\). The point \(M\) is the mid-point of \(A C\) and the points \(N\) and \(O\) lie on \(A B\) and \(B C\) such that \(M N\) and \(M O\) are arcs of circles with centres \(A\) and \(C\) respectively.
2. Show that the area of the shaded region \(B N M O\) is \(\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }\).

1. \begin{figure}[h] \end{figure} Figure 1 shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\).
Find, to 3 significant figures,

1. the length \(B C\),
2. the area of triangle \(A B C\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a design painted on the wall at a karting track. The sign consists of triangle \(A B C\) and two circular sectors of radius 2 metres and 1 metre with centres \(A\) and \(B\) respectively. Given that \(A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}\) and \(\angle A C B = 2.2\) radians,

1. use the sine rule to find the size of \(\angle A B C\) in radians to 3 significant figures,
2. show that \(\angle B A C = 0.588\) radians to 3 significant figures,
3. find the area of triangle \(A B C\),
4. find the area of the wall covered by the design.

5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

6 A motor boat can travel at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water. It is used to cross a river in which the current flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the boat makes an angle of \(60 ^ { \circ }\) to the river bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_561_1339_1692_350} The angle between the direction in which the boat is travelling relative to the water and the resultant velocity is \(\alpha\).

1. Show that \(\alpha = 16.8 ^ { \circ }\), correct to three significant figures.
2. Find the magnitude of the resultant velocity.

8 A cricket ball is hit at ground level on a horizontal surface. It initially moves at \(28 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal.

1. Find the maximum height of the ball during its flight.
2. The ball is caught when it is at a height of 2 metres above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_332_1070_1601_477} Show that the time that it takes for the ball to travel from the point where it was hit to the point where it was caught is 4.28 seconds, correct to three significant figures.
3. Find the speed of the ball when it is caught.