Revision Topic 20: Sine and Cosine Rules and 3D Trigonometry

The objectives of this unit are to:
* use the sine and cosine rules to find the length of sides and angles in triangles;
* to use the formula for the area of a triangles;
* to solve problems involving the sine and cosine rules;
* to solve problems involving trigonometry in 3 dimensions.

Brief recap of Grade B and C material:

Pythagoras’ Theorem:
This theorem, which connects the lengths of the sides in right-angled triangles, states that:

where c is the length of the hypotenuse (i.e. the side opposite the right-angle) and a and b are the lengths of the other two sides.

Note that the hypotenuse is the longest side in a right-angled triangle.

Trigonometry
The following formulae link the sides and angles in right-angled triangles:

where H is the length of the hypotenuse; O is the length of the side opposite the angle; A is the length of the side adjacent to the angle.

These formulae are often remembered using the acronym SOHCAHTOA or by using mnemonics. Here is a commonly used mnemonic: Silly Old Harry Couldn't Answer His Test On Algebra When finding angles, remember that you need to use the SHIFT key.

Further notes, examples and examination questions relating to Pythagoras’ theorem and trigonometry are contained in separate revision booklets.

Sometimes you need to calculate lengths and angles in triangles which do not contain any right-angles. This is when the sine and cosine rules are useful.

Labelling a triangle

To use the sine and cosine rules, you need to understand the convention for labelling sides and angles in any triangle.

Consider a general triangle:

Triangles are named after their vertices - the above triangle is called triangle ABC.
The three angles are commonly referred to as angles A, B and C.
The length of the sides are given lower case letters: Side a is the side opposite angle A. It is sometimes referred to as side BC. Side b is the side opposite angle B. It is equivalently called side AC. Side c is the side opposite angle C. It is also known as side AB.

A triangle doesn’t have to be labelled using the letters A, B and C.

For example, consider the triangle PQR below:

Sine Rule

The sine rule connects the length of sides and angles in any triangle ABC:

It states that: .

An alternative version of the formula is often used when finding the size of angles:

Example: Finding the length of a side

The diagram shows triangle ABC.
Calculate the length of side AB.

Solution:

To find the length of a side using the sine rule, follow these steps:

Step 1: Label the triangle using the conventions outlined earlier.
Step 2: Look to see whether any additional information can be added to the diagram (for example, can you deduce the length of any other angles?)
Step 3: Substitute information from the diagram into the sine rule formula.
Step 4: Delete the unnecessary part of the formula.
Step 5: Rearrange and then work out the length of the required side.

In our example, we begin by labelling the sides and by working out the size of the 3rd angle (using the fact that the sum of the angles in any triangle is 180°.

Substituting into the formula , we get:

As we want to calculate the length c and as the middle part of the formula is completely known, we delete the first part of the formula:

Rearranging this formula (by multiplying by sin72) gives:

i.e. c = 15.7 cm (to 1 decimal place).
Example: Finding the length of an angle

The diagram shows triangle LMN.
Calculate the size of angle LNM.

Solution:

Step 1: Label the triangle using the conventions outlined earlier.
Step 2: Substitute information from the diagram into the sine rule formula .
Step 3: Delete the unnecessary part of the formula.
Step