4.10 M) The Sine Rule – Finding an Angle
There are two different cases that we need to consider when we are working out the size of an unknown angle. The different cases depend on the size of the unknown angle. If the angle is acute (like example 1), we pretty much complete the working as you would expect. If the angle is obtuse, the working is slightly different, and we go through how answer questions like this in the second example.
The two versions of the sine rule are given below.
Angle Q is an acute angle. Find the size of angle Q to 3 significant figures.
We now move onto labelling the other side and angle. Let’s label the angle that is 48° B and the side that is opposite b (the opposite side has a length of 9).
The labelled-up triangle is given below.
We need to make sure that we do not round this answer. Your calculator should keep this as “ans” providing that you do not calculate anymore sums on the calculator.
This gives us what sinQ is. However, we want to know what Q is and not what sinQ is. Therefore, we need to get rid of sin, which we do by inversing sin. Therefore, we sin-1 both sides of the equation.
Sometimes we will be asked to find an angle in the triangle and the angle will be obtuse rather than acute. Obtuse angles are angles that are greater than 90° and less than 180°. The process for answering questions like this is exactly the same as the previous question up until the last part. I am going to explain how to answer questions like this through an example.
Angle X on the triangle below is obtuse. Find the size of angle X giving your answer to 1 decimal place.
We want to find the value of X and not sinX. Therefore, we need to get rid of the sin and we are able to do this by taking the inverse of sin from both sides; we sin-1 both sides of the equation.
From the graph above, we can see that there are 2 points of intersection and this means that there are 2 possible values for X. The first value for X will be the value that we found when we found the inverse of sin for 0.915… [sin-1(0.915…)]. This value will be 66.3° and this is shown on the graph below.
There is a slight cheat method that you can use to find the size of an obtuse angle when using the sine rule. This method involves you taking the acute angle for the angle that you are looking for off of 180°. So, for the above example, the acute angle for what we were looking for was 66.3°. Therefore, we would find the obtuse angle by taking 66.3° off of 180°.
This method is a quick method to find the answer and it is perfectly fine to use. However, it is worth trying to get your head around the longer method because you will understand where this rule comes from and by doing this, it will make it easier to remember the rule.