Sine Rule, Cosine Rule and Area of a Triangle
You are standing on the shore of Lough Corrib at a point $A$. Across a bay there are two jetties, $B$ and $C$, and you can't walk straight between them. You measure $|AB| = 300$ m, $|AC| = 450$ m and the angle between the two directions is $70^\circ$. How far apart are the jetties? There is no right angle anywhere in this triangle, so $\sin\theta = \frac{\text{opp}}{\text{hyp}}$ alone won't do. We need rules that work in any triangle. We will answer the question in the section on the cosine rule.
Labelling and the area formula
In all three formulas we use the same labelling. Each side is named with the small letter of the angle opposite it: side $a$ is opposite $\angle A$, side $b$ is opposite $\angle B$ and side $c$ is opposite $\angle C$. Side $a$ is the same as $|BC|$.
The area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. If you know two sides and the angle between them, you can get the height from trigonometry. The height of the triangle on base $b$ is $a \sin C$, so:
You will find this formula in the Formulae and Tables booklet, but you must know how to use it.
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 $ABC$, $a = 8$ cm, $b = 6$ cm and the angle $C = 30^\circ$.
Find the area of the triangle using the formula $\tfrac{1}{2}ab\sin C$.
The sine rule
The sine rule links each side to the sine of its opposite angle.
Why it works: a longer side faces a bigger angle, and the sine rule says the ratio "side ÷ sin(opposite angle)" is the same for all three sides of one triangle.
To find a side, put the unknown on top. To find an angle, turn the fractions upside down so that $\sin$ is on top:
$$\frac{\sin A}{a} = \frac{\sin B}{b}$$The ambiguous case
When you use $\sin^{-1}$, your calculator gives only an acute angle. But $\sin\theta = \sin(180^\circ - \theta)$, so there may be a second, obtuse angle with the same sine. If you are given two sides and an angle that is not between them, and the side opposite the given angle is the shorter one, there may be two possible triangles.
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 $PQR$, $\angle P = 40^\circ$, $\angle Q = 70^\circ$ and $|QR| = 9$ cm.
Use the sine rule to find $|PR|$, correct to one decimal place.
In triangle $ABC$, $|AB| = 12$ cm, $|BC| = 9$ cm and $\angle A = 40^\circ$. a) Use the sine rule to find the two possible values of the angle $C$, correct to the nearest…
In triangle $ABC$, $|AB| = 12$ cm, $|BC| = 9$ cm and $\angle A = 40^\circ$.
- a) Use the sine rule to find the two possible values of the angle $C$, correct to the nearest degree.
- b) Explain why two different triangles are possible.
- c) Find the area of each possible triangle, correct to one decimal place.
The cosine rule
The sine rule needs a matching pair (a side and its opposite angle). Sometimes you don't have one. For example, you know two sides and the angle between them (SAS), or all three sides (SSS). Then use the cosine rule. It is Pythagoras' theorem with a correction term:
Why it works: if $A = 90^\circ$, then $\cos A = 0$ and the rule becomes $a^2 = b^2 + c^2$, which is Pythagoras. For $A < 90^\circ$ the correction makes $a$ shorter; for $A > 90^\circ$, $\cos A$ is negative and $a$ becomes longer.
Now we can answer the opening question.
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.
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 triangular field $DEF$ in County Clare has sides $|DE| = 85$ m, $|EF| = 120$ m and $|DF| = 100$ m. a) Find the angle $DEF$, correct to the nearest degree. b) Find the…
A triangular field $DEF$ in County Clare has sides $|DE| = 85$ m, $|EF| = 120$ m and $|DF| = 100$ m.
- a) Find the angle $DEF$, correct to the nearest degree.
- b) Find the area of the field, correct to the nearest square metre.
- c) A farmer charges €0.40 per square metre to reseed the field. Find the cost, to the nearest euro.
Choosing the rule and solving problems
Look at what you are given and decide:
| You know… | Use… |
|---|---|
| a side and its opposite angle, plus one more side or angle | sine rule |
| two angles and any side | find the third angle ($180^\circ$ total), then sine rule |
| two sides and the angle between them | cosine rule (for the third side) or area formula |
| all three sides | cosine rule (for an angle) |
In worded problems, draw a sketch first and label the triangle.
Bearings
A bearing is measured clockwise from north and written with three digits, for example $040^\circ$ or $110^\circ$. The angle between two paths from the same point is the difference of the bearings. If this difference is more than $180^\circ$, subtract it from $360^\circ$: for example, bearings $300^\circ$ and $040^\circ$ give $360^\circ - 260^\circ = 100^\circ$.
Angles of elevation
When you walk towards a tower, the angle of elevation increases. The angle at $T$ in $\Delta ABT$ is the difference between the two elevations, because the elevation at $B$ is an exterior angle of the triangle. The angle inside the triangle at $B$ is $180^\circ$ minus the elevation at $B$.
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…
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 $52^\circ$.
- a) Find the distance $|BC|$ between the cairns, correct to the nearest metre.
- 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…
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 $54^\circ$.
- a) Show that the angle $ATB$, where $T$ is the top of the tower, is $16^\circ$.
- b) Use the sine rule to find $|BT|$, correct to one decimal place.
- c) Hence find the height of the tower, correct to the nearest metre.
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. a) Find…
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.
- a) Find the angle between their paths.
- b) Find the distance between the boats after $2$ hours, correct to one decimal place.
- c) Find the area of the triangle formed by the harbour and the two boats after $2$ hours.
The area of triangle $ABC$ is $42$ cm$^2$. It is given that $|AB| = 14$ cm, $|AC| = 8$ cm and $\angle A$ is obtuse. a) Find the size of the angle $A$, correct to the…
The area of triangle $ABC$ is $42$ cm$^2$. It is given that $|AB| = 14$ cm, $|AC| = 8$ cm and $\angle A$ is obtuse.
- a) Find the size of the angle $A$, correct to the nearest degree.
- b) Find $|BC|$, correct to one decimal place.
- c) Hence find the size of the angle $B$, correct to the nearest degree.