Updated: Sep 4, 2021
Lots of the trigonometry that students encounter when first learning the subject, such as Pythagoras’ Theorem, only really deals with right-angled triangles. This leaves a lot of students wondering about other kinds of shapes. The triangles we encounter in the real world rarely have an exact right angle, and so it seems important to be able to understand these other types as well. This is where the Sine and Cosine rules come in. These rules allow you to work out the lengths of a triangle’s side, or one of its angles, by using some information about its other sides and angles. As such, they are really useful tools which will expand your ability to make sense of shapes in own your study of Maths and Science. This post will outline what the rules are and how you can make use of them.
Sine and Cosine
Before we get started with the rules themselves, we first need to understand what Sine and Cosine are. You may have learnt about these concepts before when studying the SohCahToa rules. If you haven’t encountered them, don’t worry; all you need to know for now is that they are functions. Functions are like mathematical machines which take in numbers as inputs and produce other numbers as outputs. As Sine and Cosine are functions used for trigonometry in particular, the numbers they take as inputs tend to be angles. We can use the output numbers of the Sine and Cosine functions to help us to learn about the properties of triangles.
In order to use the Sine and Cosine functions, all you will need is a standard issue scientific calculator. On these calculators, Sine and Cosine will be represented by the ‘sin’ and ‘cos’ buttons respectively.
Sometimes you will need to take the ‘inverse’ of the Sine or Cosine function. This is represented by the symbols sin⁻¹ and cos⁻¹. You can find this on your calculator by pressing ‘shift’ and then the ‘sin’ or ‘cos’ button. The inverse works in the following way: if sin(x) = y, then sin⁻¹ (y) = x.
The Sine Rule
When using the Sine and Cosine rules, we label the triangles we are studying in a particular way. Each angle is given a capital letter, such as A, and the opposite side of that angle is given the corresponding lower-case letter, such as a.
The Sine rule is expressed by the following equation: a/sin(A) = b/sin(B) = c/sin(C)
This equation contains two equals signs, but when we use it in practice, we’ll only use two of the sets of fractions between the signs. For example, we might use the following: a/sin(A) = b/sin(B)
The Sine rule allows you to find an angle when you know the length of its opposite side, as well as another angle and the length of the opposite side of that angle. To get this you can rearrange the equation as follows: A = sin⁻¹ ((sin(B) x a) / b)
The Sine rule also allows you to find the length of a side when you know the opposite angle, as well as the length of another side and its opposite angle. To come to this, you can rearrange the equation as follows: a = (b x sin(A)) / sin(B))
The Cosine Rule
The Cosine rule is given in the following equation: a² = b² + c² - 2bc cos(A)
Like the Sine rule, the Cosine rule can be used to find either an angle or a side. You can find the length of a side if you know the other two lengths, as well as the size opposite angle of the side you’re investigating. In this case, you would rearrange the equation to make that side the subject: a = √(b² + c² - 2bc cos(A))
The Cosine also allows you to find the size of any angle of a triangle if you know the lengths of all three sides. To work this out, you’ll need to rearrange the equation as follows (remember that ‘a’ always represents the opposite side of the angle ‘A’):
A = cos⁻¹ ((b² + c² - a²) / (2 x b x c))
Now that we’ve looked at the rules, lets look at an exam style question so that we can see how to use the rules in practice:
‘Enid is in her bedroom. She starts at the door and walks in a straight line for 3m until she reaches the wall. She then turns to the right, so that there is a 60° angle between her previous direction and her new direction. She then walks in a straight line for 8m until she reaches the window. After this, she turns to the right again and walks in a straight line back to the door.
How far did Enid walk between the window and the door?’
In order to answer this question, we first need to work out which rule we’re going to apply. To do this, we first need to consider what information we’ve been given, and what information we need to find out.
Firstly, Enid’s path around her room forms a triangle, which is made up of the three straight lines she walks. The question gives you two of the sides of this triangle, and the angle between these sides. It then asks you to work out the length of the side opposite this angle. If you look back again at the equations for the two rules, it is clear that the Sine rule can’t help us very much. Instead, we’ll need to use the Cosine rule.
The next thing to notice is that, as we’re working out the length of a side, and not the size of an angle, we will need the following arrangement of the equation: a = √(b² + c² - 2bc cos(A))
As we are asked to work out the length of the line Enid walks between the window and the door, this line must be a in the equation. As such, the two lines whose lengths we are given must be b and c. The 60° angle between these two lines is therefore A.
We can plug these numbers into the equations in the following way:
a = √(3m² + 8m² - (2 x 3m x 8m x cos(60°)))
If you input this into your calculator, you’ll work out that a, which represents the length Enid’s path from the window to the door, is 7m long.
Take Things One Step at a Time
As we saw in the above example, questions which require you to use the Sine or Cosine rule can often contain a lot of information. As such, if you think you might need to use one of these rules, it can be useful to follow a few steps to make sense of the question. Firstly, you need to work out which rule to apply. You then need to find out how to arrange the equation of the rule in order to work out which bits of information you need. After this, you need to work out where the pieces of information from the question fit into your equation. By breaking the process down in this way, you can become clear about what you’re doing, and can avoid silly mistakes in the process.
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Logic Enthusiast is an independent writer and is studying for an MA in Philosophy at the University of Edinburgh. He is particularly interested in Logic and the Philosophy of Science.