Sine and Cosine Rules

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

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

1 In the triangle \(A B C , A B = 3 , B C = 4\) and angle \(A B C = 30 ^ { \circ }\). Find the following.

1. The area of the triangle.
2. The length \(A C\).
3. The angle \(A C B\).

2. In the triangle \(A B C , A B = 11 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(C A = 8 \mathrm {~cm}\).

1. Find the size of angle \(C\), giving your answer in radians to 3 significant figures.
2. Find the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to 3 significant figures.

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 \(L C\).

1. \begin{figure}[h] \end{figure} Figure 1 shows the position of three stationary fishing boats \(A , B\) and \(C\), which are assumed to be in the same horizontal plane. Boat \(A\) is 10 km due north of boat \(B\). Boat \(C\) is 8 km on a bearing of \(065 ^ { \circ }\) from boat \(B\).

1. Find the distance of boat \(C\) from boat \(A\), giving your answer to the nearest 10 metres.
2. Find the bearing of boat \(C\) from boat \(A\), giving your answer to one decimal place.

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. 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 \(x\).

\includegraphics{figure_1} Figure 1 shows triangle \(PQR\) in which \(PQ = x\), \(PR = 7 - x\), \(QR = x + 1\) and \(\angle PQR = 60°\). Using the cosine rule, find the value of \(x\). [4]

1. For the case where angle \(B A C = \frac { 1 } { 6 } \pi\) radians, find \(k\) correct to 4 significant figures.
2. For the general case in which angle \(B A C = \theta\) radians, where \(0 < \theta < \frac { 1 } { 2 } \pi\), it is given that \(\frac { \theta } { \sin \theta } > 1\). Find the set of possible values of \(k\).

2 The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

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.

1. In a triangle \(A B C\), side \(A B\) has length 10 cm , side \(A C\) has length 5 cm , and angle \(B A C = \theta\) where \(\theta\) is measured in degrees. The area of triangle \(A B C\) is \(15 \mathrm {~cm} ^ { 2 }\)
   1. Find the two possible values of \(\cos \theta\)
   Given that \(B C\) is the longest side of the triangle,
2. find the exact length of \(B C\).

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 \(B C\).

4. Figure 1 Figure 1 shows the triangle \(A B C\), with \(A B = 6 \mathrm {~cm} , B C = 4 \mathrm {~cm}\) and \(C A = 5 \mathrm {~cm}\).

1. Show that \(\cos A = \frac { 3 } { 4 }\).
2. Hence, or otherwise, find the exact value of \(\sin A\).

In this question you must show detailed reasoning. The diagram shows triangle \(ABC\). \includegraphics{figure_8} The angles \(CAB\) and \(ABC\) are each \(45°\), and angle \(ACB = 90°\). The points \(D\) and \(E\) lie on \(AC\) and \(AB\) respectively. \(AE = DE = 1\), \(DB = 2\). Angle \(BED = 90°\), angle \(EBD = 30°\) and angle \(DBC = 15°\).

1. Show that \(BC = \frac{\sqrt{2} + \sqrt{6}}{2}\). [3]
2. By considering triangle \(BCD\), show that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\). [3]

6. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle \(A B C\) in which \(A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(\angle A B C = 1.7\) radians.

1. Find the size of \(\angle A C B\) in radians. The point \(D\) lies on \(A C\) such that \(B D\) is an arc of a circle, centre \(C\).
2. Find the perimeter of the shaded region bounded by the arc \(B D\) and the straight lines \(A B\) and \(A D\).

A wooden frame is to be made to support some garden decking. The frame is to be in the shape of a sector of a circle. The sector \(OAB\) is shown in the diagram, with a wooden plank \(AC\) added to the frame for strength. \(OA\) makes an angle of \(\theta\) with \(OB\). \includegraphics{figure_2}

1. Show that the exact value of \(\sin\theta\) is \(\frac{4\sqrt{14}}{15}\) [3 marks]
2. Write down the value of \(\theta\) in radians to 3 significant figures. [1 mark]
3. Find the area of the garden that will be covered by the decking. [2 marks]

9. Diagram NOT accurately drawn \begin{figure}[h] \end{figure} Figure 3 shows the plan view of the area being used for a ball-throwing competition.
Competitors must stand within the circle \(C\) and throw a ball as far as possible into the target area, \(P Q R S\), shown shaded in Figure 3. Given that

• circle \(C\) has centre \(O\)
• \(P\) and \(S\) are points on \(C\)
• \(O P Q R S O\) is a sector of a circle with centre \(O\)
• the length of arc \(P S\) is 0.72 m
• the size of angle \(P O S\) is 0.6 radians
  1. show that \(O P = 1.2 \mathrm {~m}\)

Given also that

$$5 x ^ { 2 } + 12 x - 1500 = 0$$

Hence calculate the total perimeter of the target area, \(P Q R S\), giving your answer to the nearest metre.

A triangle \(ABC\) has sides \(AB = 6\)cm, \(BC = 11\)cm and \(AC = 13\)cm. Calculate the area of the triangle. [4]

Triangle \(ABC\) has \(AB = 8.5\) cm, \(BC = 6.2\) cm and angle \(B = 35°\). Calculate the area of the triangle. [2]

3. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.

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]{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 \(B A D\).

5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that

• \(B D = x \mathrm {~m}\)
• \(C D = ( 1 + x ) \mathrm { m }\)
• \(B C = 5 \mathrm {~m}\)
• angle \(B C D = \theta ^ { \circ }\)
  1. show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)

Given also that

\includegraphics{figure_7} Fig. 7 shows triangle ABC, with AB = 8.4 cm. D is a point on AC such that angle ADB = 79°, BD = 5.6 cm and CD = 7.8 cm. Calculate

1. angle BAD, [2]
2. the length BC. [3]

7. In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\). \(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
Calculate the possible values of the angle \(A D B\).
[0pt]

\includegraphics{figure_3} Fig. 3 Triangle ABC has AB = 9 cm, BC = 10 cm and CA = 5 cm. A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in Fig. 3.

1. Show that, to 3 decimal places, ∠BAC = 1.504 radians. [3]

Calculate,

1. the area, in cm², of the sector APQ, [2]
2. the area, in cm², of the shaded region BPQC, [3]
3. the perimeter, in cm, of the shaded region BPQC. [4]

END

\includegraphics{figure_1} The diagram shows triangle \(ABC\), with \(AC = 13.5\) cm, \(BC = 8.3\) cm and angle \(ABC = 32°\). Find angle \(CAB\). [2]

\includegraphics{figure_5} Fig. 5 shows triangle ABC, where angle ABC = \(72°\), AB = \(5.9\) cm and BC = \(8.5\) cm. Calculate the length of AC. [3]

\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).

1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
2. Hence, showing all necessary working, calculate \(\theta\). [3]

5 A triangular prism has a cross section \(A B C\) as shown in the diagram below. Angle \(A B C = 25 ^ { \circ }\) Angle \(A C B = 30 ^ { \circ }\) \(B C = 40\) millimetres. The length of the prism is 300 millimetres.
Calculate the volume of the prism, giving your answer to three significant figures.

4 The diagram shows a triangle \(A B C\) in which \(A B = 5 \mathrm {~cm} , B C = 10 \mathrm {~cm}\) and angle \(B C A = 20 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4e774e5-76fd-48ff-9bce-a995b3ba517b-2_355_767_1695_689}

1. Find angle \(B A C\), given that it is obtuse.
2. Find the shortest distance from \(A\) to \(B C\).

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 \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).

In a triangle \(ABC\), \(AB = 9\) cm, \(BC = 7\) cm and \(AC = 4\) cm.

1. Show that \(\cos CAB = \frac{2}{3}\). [2]
2. Hence find the exact value of \(\sin CAB\). [2]
3. Find the exact area of triangle \(ABC\). [2]

A parallelogram has sides of length 6 cm and 4.5 cm. The larger interior angles of the parallelogram have size \(\alpha\) Given that the area of the parallelogram is 24 cm², find the exact value of \(\tan \alpha\) [4 marks]

\includegraphics{figure_6} The diagram shows triangle \(ABC\), in which \(AB = 3\) cm, \(AC = 5\) cm and angle \(ABC = 2.1\) radians. Calculate

1. angle \(ACB\), giving your answer in radians, [2]
2. the area of the triangle. [3]

An arc of a circle with centre \(A\) and radius 3 cm is drawn, cutting \(AC\) at the point \(D\).

1. Calculate the perimeter and the area of the sector \(ABD\). [4]