Back to AQA Trigonometry (H) Home
4.10 Q) 3D Trigonometry
4.10 Q) 3D Trigonometry
Sometimes in the exam you will be asked trigonometry questions to do with 3D shapes. We answer questions like this by creating right angle triangles within the 3D shape.
Before you work through this section, make sure that you are comfortable with using trigonometry to work out the length of a side and the size of an angle.
The three trigonometry formula triangles are shown below:
Before you work through this section, make sure that you are comfortable with using trigonometry to work out the length of a side and the size of an angle.
The three trigonometry formula triangles are shown below:
Let’s have an example.
Example 1
The shape below is a rectangle. All of the measurements are in cm.
The shape below is a rectangle. All of the measurements are in cm.
Find the following:
- Find the length of AC. Give your answer to one decimal place.
- Find the size of angle CAG. Give your answer to the nearest degree.
Part 1
The first part of this question asks us to work out the length of AC. We can work out the length of AC by using the right angle triangle ABC. The triangle ABC is shown in green on the 3D diagram below.
The first part of this question asks us to work out the length of AC. We can work out the length of AC by using the right angle triangle ABC. The triangle ABC is shown in green on the 3D diagram below.
It is best to draw the right angle triangle that we are using out seperately and in 2D form. The triangle ABC is shown below:
We are looking for the length of AC and I have labelled this as x on the above diagram. We are going to be using trigonometry to work out the length of AC. Therefore, we need to label the sides in the triangle to find which trigonometry formula triangle we are going to use. For the triangle ABC, we have an angle and the adjacent, and we are looking for the hypotenuse. The labelled up triangle is shown below:
The formula triangle that contains the adjacent and hypotenuse is the CAH formula triangle.
We are looking for the hypotenuse, so we cover H up on the formula triangle above, which tells us that we find the hypotenuse by dividing the adjacent by cos of the angle.
We now sub the values in for the adjacent and the angle.
The next step is to round the answer to 1 decimal place.
Therefore, the length of AC is 11.8 cm.
Part 2
The second part of this question asks us to find the size of the angle CAG. We are able to find the size of this angle by using the right angle triangle CAG. I have drawn this triangle in orange on the 3D diagram below.
The second part of this question asks us to find the size of the angle CAG. We are able to find the size of this angle by using the right angle triangle CAG. I have drawn this triangle in orange on the 3D diagram below.
Like with the previous question, it is best to draw out a separate 2D version of this triangle and this triangle is drawn below.
The angle that we are looking for is CAG and I have labelled this angle y. Also, the length of AC is 11.8 cm (we found this in the previous part).
The next step is to label the triangle up and determine what trigonometry formula triangle we need to use. For the triangle below, we have the adjacent (the side that is 11.8 cm) and the opposite (the side that is 6 cm).
The next step is to label the triangle up and determine what trigonometry formula triangle we need to use. For the triangle below, we have the adjacent (the side that is 11.8 cm) and the opposite (the side that is 6 cm).
The trigonometry formula triangle that contains the adjacent and the opposite is TOA.
We are looking for the angle, so we find the calculation that we undertake by covering up T in the formula above.
We now sub the values into the formula.
We want to find the value y and not tan(y). Therefore, we need to get out of tan and we do this by taking the inverse of tan from both sides of the equation; we tan-1 both sides of the equation.
The question asks us to round our answer to the nearest degree.
Therefore, angle CAG is 27°.