Back to AQA Trigonometry (H) Home
4.10 E) Finding a Side
4.10 E) Finding a Side
We will be using trigonometry to either find the length of a side or the size of an angle. We are going to be looking at finding the length of a side in this section. In the next section, we will be looking at find an unknown angle.
The first step in finding the length of a side is to label up your triangle; we only want to label up the sides that we are given or are looking for. After we have labelled up the triangle, we then find the trigonometry formula triangle that contains the two sides that we have labelled. The next step is to cover up the component in the formula triangle that we are looking for. The covering up of what we are looking for gives us the calculation that we need to undertake to find what we are looking for. We then sub the values in and complete the calculation to obtain our answer.
This will all make a lot more sense after we have gone through some examples.
The trigonometry formula triangles are shown below.
The first step in finding the length of a side is to label up your triangle; we only want to label up the sides that we are given or are looking for. After we have labelled up the triangle, we then find the trigonometry formula triangle that contains the two sides that we have labelled. The next step is to cover up the component in the formula triangle that we are looking for. The covering up of what we are looking for gives us the calculation that we need to undertake to find what we are looking for. We then sub the values in and complete the calculation to obtain our answer.
This will all make a lot more sense after we have gone through some examples.
The trigonometry formula triangles are shown below.
Example 1
What is the length of x in the triangle below? Give you answer to 3 significant figures.
What is the length of x in the triangle below? Give you answer to 3 significant figures.
The first step in answering this question is to label the sides to see which of the formula triangles we will be using. In the above right-angled triangle, we are given/ are looking for an angle, the opposite and the hypotenuse. The labelled triangle is shown below.
We are looking for the formula triangle that contains the opposite and the hypotenuse. This means that we will be using the SOH triangle.
The question asks us to find the opposite. We find the calculation that we undertake to find the opposite by covering up O in the above formula triangle. This tells us that we need to multiply S by H. S stands for sin of the angle and H stands for the hypotenuse.
We are asked to give the answer to 3 significant figures. We count the significant figures starting from the left with the first non-zero number. We then place a line after the 3rd significant figure. If the number to the right of this line is 5 or greater, we round up. If the number is less than 5, we round down.
The number to the right of the line is 5, which means that we round up. Rounding up involves increasing the number to the left of the red line by 1. Therefore, x is 5.74 cm to 3 significant figures.
Example 2
What is the length of y? Give your answer to 2 decimal places.
What is the length of y? Give your answer to 2 decimal places.
Like before, the first step is to see what formula triangle we need to use. We do this by labelling everything that we are given in the question/ asked to find. In the above triangle, we are given the opposite and we want to find the adjacent.
This means that we are looking for the formula triangle that contains the opposite and adjacent, which is the TOA formula triangle.
The next step is to find the operation that we need to undertake and we do this by covering up what we are trying to find. As we are wanting to find the adjacent, we cover A up in the formula triangle. This tells us that we need to divide O by T. O is the opposite and T is tan of the angle.
The question asks us to give our answer to 2 decimal places. Therefore, we place a line after the second decimal place and look at the number that is to the right of this line.
The number to the right of the red line is an 8, which means that we round up. This results in the number to the left of the red line increasing by 1. Therefore, y to 2 decimal places is 8.58 cm.