Following on from the previous blog on Pythagoras’ theorem, this blog will be a guide to solving trigonometry questions. This guide will show you how to use trigonometry to find a missing side in a right-angled triangle. The next blog will build on this idea and show you how to find a missing angle in a right-angled triangle.
Trigonometry is examinable on both the foundation and higher GCSE. These questions are more likely to come up in calculator exams, however especially in the higher tier they could come up in a non-calculator exam where you may also be examined on common sin, cos and tan values.
In the same vain as Pythagoras’ theorem, you can only do trigonometry on right-angled triangles. It is also important to note that you can only use trigonometry if you have an angle and two sides labelled.
Firstly, you need to remember an acronym – SOHCAHTOA.
Here are what the letters mean:
S = sin
C = cos
T = tan
O = opposite
A = adjacent
H = hypotenuse
Sin, cos and tan are the three trigonometric functions. You can find them on your calculator – see below.
In a right-angled triangle, the hypotenuse is always the longest side. It is always the side opposite the right angle.
The opposite is always the side opposite the labelled angle (note: the labelled angle is not the right angle).
The adjacent is the other side – this is the side next to the angle.
See the two examples below, in both cases the labelled angle is x.
SOHCAHTOA is an acronym to for the following trigonometric formulas.
Sin = Opposite ÷ Hypotenuse (SOH)
Cos = Adjacent ÷ Hypotenuse (CAH)
Tan = Opposite ÷ Adjacent (TOA)
I encourage my students to write SOHCAHTOA in the following way:
In the same way as the distance-speed-time, density-mass-volume and pressure-force-area triangles discussed in a previous blog, these triangles are a quicker and easier way of remembering the trigonometric formulas. Technically there are three different configurations of each triangle you need to know, which is 9 formulas – and 3 triangles are much easier to remember than 9 formulas.
Here’s a quick reminder of how to use the triangles.
Say I was using the SOH triangle, and I wanted to find the Hypotenuse.
I put my hand over the H, to leave me with O ÷ S.
So,
Hypotenuse = Opposite ÷ Sin
This can be used for all three triangles.
So now we know what SOHCAHTOA stands for and how to use it. Now let’s work through an example and follow the steps to solve it.
Example 1
Find x.
Step 1 – Label the triangle
We need to label the sides of the triangle with H (hypotenuse), O (opposite) and A (adjacent).
H is the side opposite the right angle, O is the side opposite the angle and A is other side.
Step 2 – Pick one of the SOHCAHTOA triangles.
This is usually the step that trips people up. We now need to choose one of the three SOHCAHTOA triangles below.
To do this, we look at the information we care given in the question and what we need to work out. This is usually the step that students find difficult.
In this question, we are given the adjacent (6.3 cm) and are being asked to find the hypotenuse (as this is the side labelled x). So we want to pick the triangle which has A and H in it.
You will see that the only SOHCAHTOA triangle which has both A and H in it is CAH. So this is the triangle we are going to use.
Step 3 – Use the chosen triangle to solve.
Now we can use the trick with the CAH triangle to find the missing side.
In this question, we are being asked to find H (as this is the side labelled x) so we cover up the H.
This leaves us with Hypotenuse = Adjacent ÷ Cos.
Now, you will notice on your calculator that when you press cos a bracket immediately appears after it. This is because sin, cos and tan are trigonometric functions which need an input. The input is the angle given in the question (34°). So our final calculation will be:
x = 6.3 ÷ cos(34)
= 7.6 cm (to 1 decimal place)
Example 2
Find x.
First, label the triangle.
Now pick one of SOH, CAH and TOA.
We are asked to find O and we are given H, so we want the triangle that has O and H in it. This is SOH.
We are being asked to find O, so we cover up the O:
This leaves us with
O = S × H
x = sin(57) × 14.5
= 12.2 cm (to 1 decimal place)
One final example.
Example 3
Find x.
Label the triangle:
We are asked to find A and we are given O, so we want the triangle that has O and A in it. This is TOA.
We are asked to find A, so we cover up the A:
This leaves us with:
A = O ÷ T
x = 7.1 ÷ tan(42)
= 7.9 cm (to 1 decimal place)
So there is the method for using trigonometry to find a missing side of a right-angled triangle. You can work on some more trigonometry questions on our worksheet. The next blog will build on this and show you how to use trigonometry to find the missing angle of a right-angled triangle.
If you need someone to show you this method in person, book a free taster session.