# Non right angled triangles

For right-angled triangles, we have Pythagoras’ Theorem and SOHCAHTOA. However, these methods do not work for non-right angled triangles. For non-right angled triangles, we have the cosine rule, the sine rule and a new expression for finding area.

In order to use these rules, we require a technique for labelling the sides and angles of the non-right angled triangle. This may mean that a relabelling of the features given in the actual question is needed. See the non-right angled triangle given here. Angle A is opposite side a, angle B is opposite side B and angle C is opposite side c. We determine the best choice by which formula you remember in the case of the cosine rule and what information is given in the question but you must always have the UPPER CASE angle OPPOSITE the LOWER CASE side.

## The Cosine Rule

These formulae represent the cosine rule. Note that it is not necessary to memorise all of them – one will suffice, since a relabelling of the angles and sides will give you the others. Students tend to memorise the bottom one as it is the one that looks most like Pythagoras.

We use the cosine rule to find a missing side when all sides and an angle are involved in the question. It may also be used to find a missing angle if all the sides of a non-right angled triangle are known. See Examples 1 and 2.

These formulae represent the cosine rule. Note that it is not necessary to memorise all of them – one will suffice, since a relabelling of the angles and sides will give you the others. Students tend to memorise the bottom one as it is the one that looks most like Pythagoras.

We use the cosine rule to find a missing side when all sides and an angle are involved in the question. It may also be used to find a missing angle if all the sides of a non-right angled triangle are known. See Examples 1 and 2.

## The Sine Rule

This formula represents the sine rule. The sine rule can be used to find a missing angle or a missing side when two corresponding pairs of angles and sides are involved in the question. This is different to the cosine rule since two angles are involved. This is a good indicator to use the sine rule in a question rather than the cosine rule. See Example 3.

Note that when using the sine rule, it is sometimes possible to get two answers for a given angleside length, both of which are valid. See Example 4.

or

## The Area of a Non-Right Angled Triangle

These formulae represent the area of a non-right angled triangle. Again, it is not necessary to memorise them all – one will suffice (see Example 2 for relabelling). It is the analogue of a half base times height for non-right angled triangles.

Note that to maintain accuracy, store values on your calculator and leave rounding until the end of the question. You can round when jotting down working but you should retain accuracy throughout calculations. See Examples 5 and 6.

## Examples

Find the length of the side marked x in the following triangle:

Solution:

### (Cosine Rule)

Find x using the cosine rule according to the labels in the triangle above. Firstly, choose , , and so . Then, substitute into the cosine rule:
.

It follows that x=4.87 to 2 decimal places.

The triangle PQR has sides cm, cm and cm. Angle is . Find the value of .

Solution:

Begin by drawing the triangle. The cosine rule says that

noting that the little given in the question might be different to the little in the formula.

In this example, we require a relabelling and so we can create a new triangle where we can use the formula and the labels that we are used to using. Answering the question given amounts to finding side a in this new triangle. The formula gives

.

The trick is to recognise this as a quadratic in and simplifying to

Using the quadratic formula, the solutions of this equation are and to 2 decimal places. Click here to find out more on solving quadratics. Since a must be positive, the value of c in the original question is 4.54 cm.

Find the angle marked in the following triangle to 3 decimal places:

Solution:

### (Sine rule)

This time, find using the sine rule according to the labels in the triangle above. Firstly, choose , and so and . Then use one of the equations in the first equation for the sine rule:

.
It follows that

to 3 decimal places.

Note how much accuracy is retained throughout this calculation. If we rounded earlier and used 4.699 in the calculations, the final result would have been x=26.545 to 3 decimal places and this is incorrect.

In triangle , length m, length m and the angle at is . Sketch the two possibilities for this triangle and find the two possible values of the angle at to 2 decimal places.

Solution:

The two possible triangles are given by:

The sine rule will give us the two possibilities for the angle at , this time using the second equation for the sine rule above:

Solving gives or . See more on solving trigonometric equations. It follows that the two values for , found using the fact that angles in a triangle add up to 180, are and to 2 decimal places

Find the area of this triangle.

Solution:

### (Area)

It is not necessary to find in this example as the area of this triangle can easily be found by substituting , and into the formula for the area of a triangle. Hence,

square units to 2 decimal places.

Find the area of the triangle with sides 22km, 36km and 47km to 1 decimal place.

Solution:

To find the area of this triangle, we require one of the angles. An angle can be found using the cosine rule choosing , and :

or

Simplifying gives and so . It follows that the area is given by

.

## Videos

Using a trigonometric identity to determine a cos value which is used in an application of the cosine rule.

Using the sine and cosine rule to find angles and lengths in a quadrilateral flowerbed.

## Extra Resources

### Questions by Topic

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