Sine rule, cosine rule and area of a triangle – Year 11 GCSE maths worksheet with answers to download as PDF

In triangle \(ABC\), side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\) and side \(c\) is opposite angle \(C\). The sine rule is \(\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}\).

1. \(A=40^\circ\), \(B=65^\circ\), \(a=8\) cm. Find \(b\) to 2 decimal places. [3 marks]
2. Find angle \(C\). [1 mark]
3. Find side \(c\) to 2 decimal places. [3 marks]

In triangle \(ABC\), \(a=9\) cm, \(b=12\) cm and angle \(A=35^\circ\).

1. Use the sine rule to find acute angle \(B\) to 1 decimal place. [3 marks]
2. Explain why angle \(B\) could also be obtuse and find that value. [2 marks]
3. Which angle \(B\) is correct if the triangle is known to be acute? [1 mark]

The cosine rule is \(a^2=b^2+c^2-2bc\cos A\).

1. In triangle \(ABC\), \(b=7\) cm, \(c=9\) cm and \(A=48^\circ\). Find \(a\) to 2 decimal places. [3 marks]
2. Find angle \(B\) using the sine rule. [3 marks]
3. Check that the three angles add to \(180^\circ\) after finding \(C\). [2 marks]

A triangle has sides \(a=7\) cm, \(b=8\) cm and \(c=10\) cm.

1. Write the cosine rule in the form for finding \(\cos C\). [1 mark]
2. Find the largest angle \(C\) to 1 decimal place. [3 marks]
3. Find angle \(A\) and angle \(B\). [3 marks]

The area of a triangle is \(\tfrac12ab\sin C\).

1. Two sides are 8 cm and 11 cm with the included angle \(37^\circ\). Find the area to 1 decimal place. [2 marks]
2. Another triangle has sides 5 cm and 6 cm and area 12 cm². Find the included angle, acute. [3 marks]
3. Explain why \(\tfrac12\times5\times6\times\sin C\) cannot give an area greater than 15 cm². [1 mark]

Use exact values of sine without a calculator.

1. A triangle has sides 4 cm and 6 cm with an included angle of \(30^\circ\). Find its exact area. [2 marks]
2. The included angle is now \(60^\circ\). Find the exact area. [2 marks]
3. Find the area for the included angle \(90^\circ\) and explain which angle gives the largest area. [2 marks]

Two ships leave a port \(P\). Ship \(A\) sails 12 km on a bearing of \(050^\circ\). Ship \(B\) sails 9 km on a bearing of \(110^\circ\).

1. Show that angle \(APB=60^\circ\). [1 mark]
2. Find the distance \(AB\). [3 marks]
3. Find the area of triangle \(APB\). [2 marks]
4. Find angle \(PAB\) to 1 decimal place. [3 marks]

A field is a triangle with two sides of 120 m and 150 m and an angle of \(52^\circ\) between them.

1. Find the area of the field to the nearest square metre. [2 marks]
2. Find the length of the third side. [3 marks]
3. Fencing costs £12.50 per metre. Find the cost of fencing all the way round. [3 marks]

Points \(A\) and \(B\) are 200 m apart on level ground. A tower \(T\) is seen from \(A\) at an angle \(TAB=62^\circ\) and from \(B\) at \(TBA=71^\circ\).

1. Find angle \(ATB\). [1 mark]
2. Find the distance \(AT\) to 1 decimal place. [3 marks]
3. Find the distance \(BT\). [2 marks]
4. From \(A\) the angle of elevation of the top of the tower is \(12^\circ\). Find the height of the tower. [3 marks]

A triangular sail has sides 3.2 m, 4.1 m and 5.0 m.

1. Find the largest angle to 1 decimal place. [3 marks]
2. Find the area of the sail to 2 decimal places. [3 marks]
3. Sail cloth costs £18 per m². Find the cost of the cloth for 6 sails. [3 marks]

A plane flies 250 km on a bearing of \(030^\circ\) and then 180 km on a bearing of \(100^\circ\).

1. Show that the angle between the two legs, inside the triangle, is \(110^\circ\). [2 marks]
2. Find the direct distance from the start. [3 marks]
3. Find the bearing of the end point from the start to the nearest degree. [4 marks]

To find the distance across a lake, a surveyor stands at \(C\) and measures \(CA=85\) m, \(CB=120\) m and angle \(ACB=74^\circ\).

1. Find \(AB\) to 1 decimal place. [3 marks]
2. Find angle \(CAB\). [3 marks]
3. Find the area of triangle \(ABC\). [2 marks]

A triangle has sides 5 cm, 7 cm and 10 cm.

1. Use the cosine rule to find the largest angle. [3 marks]
2. Explain how the sign of \(\cos C\) shows that the angle is obtuse. [1 mark]
3. Find the other two angles. [4 marks]

A triangle has sides 10 cm and 8 cm and an area of 30 cm².

1. Show that \(\sin C=0.75\). [2 marks]
2. Find the two possible sizes of the included angle \(C\). [3 marks]
3. Find the third side for each case. [4 marks]

In quadrilateral \(ABCD\): \(AB=6\) cm, \(BC=7\) cm, angle \(ABC=80^\circ\), \(CD=5\) cm and \(DA=9\) cm.

1. Find \(AC\) to 2 decimal places. [3 marks]
2. Find angle \(ACD\). [3 marks]
3. Find the area of the quadrilateral. [5 marks]

In triangle \(ABC\), \(A=45^\circ\), \(B=60^\circ\) and \(a=6\) cm.

1. Use the sine rule to show that \(b=3\sqrt6\). [4 marks]
2. Find angle \(C\) and the exact area using \(\tfrac12ab\sin C\) (leave in surd form, \(\sin75^\circ\) left as it is). [3 marks]
3. Find \(b\) to 2 decimal places. [1 mark]

A triangle has sides 5 cm and 8 cm with an included angle of \(120^\circ\).

1. Show that the third side squared is 129. [3 marks]
2. Find the third side. [1 mark]
3. Find the area exactly. [3 marks]
4. Find the smallest angle. [3 marks]

A triangle has sides 7 cm, 8 cm and 13 cm.

1. Show that the largest angle is exactly \(120^\circ\). [4 marks]
2. Find the exact area. [3 marks]
3. Find the height from the 13 cm side to the opposite vertex. [2 marks]

Lighthouses \(A\) and \(B\) are 10 km apart with \(B\) due east of \(A\). A ship \(S\) is on a bearing of \(030^\circ\) from \(A\) and a bearing of \(310^\circ\) from \(B\).

1. Show that angle \(SAB=60^\circ\) and angle \(SBA=40^\circ\). [3 marks]
2. Find angle \(ASB\). [1 mark]
3. Find the distance \(AS\). [3 marks]
4. Find how far the ship is north of the line \(AB\). [2 marks]

In triangle \(ABC\), \(CD\) is the perpendicular from \(C\) to \(AB\).

1. Write \(CD\) in terms of \(a\) and \(\sin B\). [2 marks]
2. Hence show that the area of the triangle is \(\tfrac12ac\sin B\). [2 marks]
3. The area can also be written as \(\tfrac12bc\sin A\). Equate the two expressions and show that \(\dfrac a{\sin A}=\dfrac b{\sin B}\). [4 marks]