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4.10 L) The Sine Rule – Finding a Length
4.10 L) The Sine Rule – Finding a Length
We are able to use the sin rule to work out the size of an angle or the length of a side in a triangle. We can use the sin rule for any triangle; the triangle does not have to be a right-angle triangle. The triangle below is the triangle that I am going to use to explain the sin rule.
When we are using the sin rule, we need to make sure that we label the triangle up correctly. I am going to label the sides with lower case letters and angles with capital letters. I am going to label my sides a, b and c. The angle that is opposite the side will be labelled with the same letter, but in capital form.
The sin rule is given below:
The sin rule is given below:
We can rearrange the sin rule to give what is shown below (we flip the sin rule in the form above; the numerators become the denominators, and the denominators become the numerators):
It does not matter what version of the sine rule that you use providing that you make sure that all of the sides are the numerators and all of the angles are the denominators (or vice versa). You must never have a mixture of sides as numerators and denominators.
The first version is usually used to calculate the length of an unknown side and the second version is usually used to calculate the size of an unknown angle; it is easier to have whatever you are looking for as a numerator rather than a denominator.
Let’s have a few examples.
The first version is usually used to calculate the length of an unknown side and the second version is usually used to calculate the size of an unknown angle; it is easier to have whatever you are looking for as a numerator rather than a denominator.
Let’s have a few examples.
Example 1
What is the value of x in the triangle below? Give you answer to 1 decimal place.
What is the value of x in the triangle below? Give you answer to 1 decimal place.
The first step in answering this question is label all of the sides and angles up that we need or want to find. The question is asking us to find the length of one of the sides and it makes sense to label this side a (it will be a lower-case a as it is a side and not an angle). We will label the angle that is opposite this side A (it is a capital because it is an angle). Therefore, the angle that is 60° will be labelled as A.
I am going to label the side that we are given b (the side that is 10) and the angle that is opposite B (the angle that is 60°). The labelled triangle is shown below.
I am going to label the side that we are given b (the side that is 10) and the angle that is opposite B (the angle that is 60°). The labelled triangle is shown below.
We now use the sine rule and sub the values into their appropriate places. As we are finding a length, it is best to use the formula that has the lengths as numerators.
We want to the find the value of x, so we multiply both sides of the equation by sin(60).
The question asks us to give this answer to 1 decimal place. We do this by placing a line after the first decimal place and looking at the number that is to the right of the red line.
The number to the right of the red line is 4, which is less than 5. Therefore, we round our answer down, which means that x is 12.2 units to 1 decimal place.