Exercises: Sine Rule, Cosine Rule and Area of a Triangle

In triangle $ABC$, $a = 8$ cm, $b = 6$ cm and the angle $C = 30^\circ$.

Find the area of the triangle using the formula $\tfrac{1}{2}ab\sin C$.

In triangle $PQR$, $\angle P = 40^\circ$, $\angle Q = 70^\circ$ and $|QR| = 9$ cm.

Use the sine rule to find $|PR|$, correct to one decimal place.

In triangle $XYZ$, $|XY| = 7$ cm, $|XZ| = 10$ cm and $\angle X = 60^\circ$.

Use the cosine rule to find $|YZ|$, correct to one decimal place.

A hill-walker stands at point $A$ and sees two cairns, $B$ and $C$. The cairn $B$ is $450$ m from $A$ and the cairn $C$ is $620$ m from $A$. The angle $BAC$ is $52^\circ$.

1. a) Find the distance $|BC|$ between the cairns, correct to the nearest metre.

1. b) Find the area of triangle $ABC$, correct to the nearest square metre.

From a point $A$ on level ground, the angle of elevation of the top of the Spire-style tower is $38^\circ$. The student walks $50$ m directly towards the tower to a point $B$ and the angle of elevation is now $54^\circ$.

1. a) Show that the angle $ATB$, where $T$ is the top of the tower, is $16^\circ$.

1. b) Use the sine rule to find $|BT|$, correct to one decimal place.

1. c) Hence find the height of the tower, correct to the nearest metre.

A triangular field $DEF$ in County Clare has sides $|DE| = 85$ m, $|EF| = 120$ m and $|DF| = 100$ m.

1. a) Find the angle $DEF$, correct to the nearest degree.

1. b) Find the area of the field, correct to the nearest square metre.

1. c) A farmer charges €0.40 per square metre to reseed the field. Find the cost, to the nearest euro.

Two boats leave Howth harbour at the same time. Boat $A$ sails on a bearing of $040^\circ$ at $18$ km/h. Boat $B$ sails on a bearing of $110^\circ$ at $24$ km/h.

1. a) Find the angle between their paths.

1. b) Find the distance between the boats after $2$ hours, correct to one decimal place.

1. c) Find the area of the triangle formed by the harbour and the two boats after $2$ hours.

In triangle $ABC$, $|AB| = 12$ cm, $|BC| = 9$ cm and $\angle A = 40^\circ$.

1. a) Use the sine rule to find the two possible values of the angle $C$, correct to the nearest degree.

1. b) Explain why two different triangles are possible.

1. c) Find the area of each possible triangle, correct to one decimal place.

The area of triangle $ABC$ is $42$ cm$^2$. It is given that $|AB| = 14$ cm, $|AC| = 8$ cm and $\angle A$ is obtuse.

1. a) Find the size of the angle $A$, correct to the nearest degree.

1. b) Find $|BC|$, correct to one decimal place.

1. c) Hence find the size of the angle $B$, correct to the nearest degree.