1.05b Sine and cosine rules: including ambiguous case

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )

The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\).
3. Show that the size of the angle \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
5. find the two possible position vectors of point \(D\).
   

5. \begin{figure}[h] \end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at a point \(O\). The force \(\mathbf { P }\) has magnitude 15 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The angle between \(\mathbf { P }\) and \(\mathbf { Q }\) is \(150 ^ { \circ }\), as shown in Figure 1. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(\mathbf { R }\). Given that the angle between \(\mathbf { R }\) and \(\mathbf { Q }\) is \(50 ^ { \circ }\), find

1. the magnitude of \(\mathbf { R }\),
2. the value of \(X\).

6. \begin{figure}[h] \end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at \(O\). The angle between the lines of action of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(120 ^ { \circ }\) as shown in Figure 4. The force \(\mathbf { P }\) has magnitude 20 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\). Given that the magnitude of \(\mathbf { R }\) is \(3 X\) newtons, find, giving your answers to 3 significant figures

1. the value of \(X\),
2. the magnitude of \(( \mathbf { P } - \mathbf { Q } )\).

7. \begin{figure}[h] \end{figure} Two forces, \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle. The force \(\mathbf { P }\) has magnitude 8 N and the force \(\mathbf { Q }\) has magnitude 5 N . The angle between the directions of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(50 ^ { \circ }\), as shown in Figure 3. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\).

1. Find, to 3 significant figures, the magnitude of \(\mathbf { R }\).
2. Find, to the nearest degree, the size of the angle between the direction of \(\mathbf { P }\) and the direction of \(\mathbf { R }\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac { 4 } { 3 }\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,

1. find the value of \(m\),
2. find the magnitude of the force exerted on the pulley by the string,
3. find the direction of the force exerted on the pulley by the string. \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-21_2258_50_314_37}
   

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

   \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-23_2258_50_314_37}
   
   \includegraphics[max width=\textwidth, alt={}]{6ab8838f-d6f8-4761-8def-1022d97d4e82-24_2655_1830_105_121}

   VIAV SIHI NI JIIYM IONOO

   VI4V SIHI NI IIIIMM ION OO

   VEYV SIHI NI JLIYM ION OC

3 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_488_604_895_769} A landmark \(L\) is observed by a surveyor from three points \(A , B\) and \(C\) on a straight horizontal road, where \(A B = B C = 200 \mathrm {~m}\). Angles \(L A B\) and \(L B A\) are \(65 ^ { \circ }\) and \(80 ^ { \circ }\) respectively (see diagram). Calculate

1. the shortest distance from \(L\) to the road,
2. the distance \(L C\).

2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate

1. the area of the triangle,
2. the length of \(C A\),
3. the size of angle \(C\).

4 \includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-2_515_713_1567_715} In the diagram, angle \(B D C = 50 ^ { \circ }\) and angle \(B C D = 62 ^ { \circ }\). It is given that \(A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}\) and \(B C = 16 \mathrm {~cm}\).

1. Find the length of \(B D\).
2. Find angle \(B A D\).

4 \includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-3_309_1084_269_532} In the diagram, \(A B C D\) is a quadrilateral in which \(A D\) is parallel to \(B C\). It is given that \(A B = 9 , B C = 6\), \(C A = 5\) and \(C D = 15\).

1. Show that \(\cos B C A = - \frac { 1 } { 3 }\), and hence find the value of \(\sin B C A\).
2. Find the angle \(A D C\) correct to the nearest \(0.1 ^ { \circ }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-3_476_1018_1000_566} The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the arc \(D C\).

1. Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
2. Find the area of the shaded segment.
3. Find the perimeter of the shaded segment.

4 Fig. 4 For triangle ABC shown in Fig. 4, calculate

1. the length of BC ,
2. the area of triangle ABC .

11 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(A B , B C\) and \(C A\). The road AC runs due North and the measurements shown are in metres. \begin{figure}[h] \end{figure}

1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of the village green. The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_440_1002_1436_737} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} The new road is an arc of a circle with centre O and radius 130 m .
3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
   (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures.

11

1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_661_869_404_680} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} (A) Calculate the total length of the course for this race.
   (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC.
2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_709_821_1603_703} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} Calculate the total length of the course for this race.

10 Arrowline Enterprises is considering two possible logos: \begin{figure}[h] \end{figure} Fig. 10.2

1. Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC . Find the length of AB and hence find the area of this logo.
2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that \(\mathrm { ST } = 16.2 \mathrm {~cm}\) to 3 significant figures.
   Find the area and perimeter of this logo.

8 In the triangle \(\mathrm { ABC } , \mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 6 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 110 ^ { \circ }\).
Find the length of the side BC .

10 At 1200 the captain of a ship observes that the bearing of a lighthouse is \(340 ^ { \circ }\). His position is at A.
At 1230 he takes another bearing of the lighthouse and finds it to be \(030 ^ { \circ }\). During this time the ship moves on a constant course of \(280 ^ { \circ }\) to the point B . His plot on the chart is as shown in Fig. 11 below. \begin{figure}[h] \end{figure}

1. Write down the size of the angles LAB and LBA .
2. The captain believes that at A he is 5 km from L . Assuming that LA is exactly 5 km , show that LB is 4.61 km , correct to 2 decimal places, and find AB . Hence calculate the speed of the ship.
3. The speed of the ship is actually 10 kilometres per hour. Given that the bearings of \(340 ^ { \circ }\) and \(030 ^ { \circ }\) and the ship's course of \(280 ^ { \circ }\) are all accurate, calculate the true value of the distance LA.

8 In the triangle ABC shown, \(\mathrm { AB } = 8 \mathrm {~cm}\). \(\mathrm { AC } = 12 \mathrm {~cm}\) and angle \(\mathrm { ABC } = 82 ^ { \circ }\). Find \(\theta\) correct to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-3_382_540_1492_718}

6 A and B are points on the same side of a straight river. A and B are 180 metres apart. The angles made with a jetty J on the opposite side of the river \(78 ^ { \circ }\) and \(56 ^ { \circ }\) respectively as shown. \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_332_681_1451_565} Not to scale Calculate the width of the river correct to the nearest metre.

7. The diagram 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. Show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
2. Find the size of \(\angle B C D\) in degrees to 1 decimal place.
3. Find the area of quadrilateral \(A B C D\).

2. The diagram shows triangle \(P Q R\) in which \(P Q = x , P R = 7 - x , Q R = x + 1\) and \(\angle P Q R = 60 ^ { \circ }\). Using the cosine rule, find the value of \(x\).

2. The diagram 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

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

7. \includegraphics[max width=\textwidth, alt={}, center]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-3_376_892_221_427} The diagram 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. 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.

4. The diagram 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.

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_765_757_203_764} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045 ^ { \circ }\). R is 9.2 km from P on a bearing of \(113 ^ { \circ }\), so that angle QPR is \(68 ^ { \circ }\). Calculate the distance and bearing of R from Q .
2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_527_1474_1452_404} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} BC is an arc of a circle with centre A and radius 80 cm . Angle \(\mathrm { CAB } = \frac { 2 \pi } { 3 }\) radians.
   EC is an arc of a circle with centre D and radius \(r \mathrm {~cm}\). Angle CDE is a right angle.
   1. Calculate the area of sector ABC .
   2. Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
   3. Hence calculate the area of cross-section of the rudder. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-5_695_1012_271_600} \captionsetup{labelformat=empty} \caption{Fig. 12}
      \end{figure} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
      1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough.
      2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15\), for \(0 \leqslant x \leqslant 0.5\). Calculate \(\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x\) and state what this represents.
         Hence find the volume of water in the trough as given by this model.

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.