scientific calculator ii
Foundation GCSE, Higher GCSE

A guide to trigonometry (SOHCAHTOA) – Part 2

In the last blog, I introduced a method for using trigonometry to find the missing side in a right-angled triangle. This blog will build on that and show you how to find the missing angle in a right-angled triangle. I would recommend reading that before this one for an introduction to trigonometry.

Example 1

Find x.

right angled triangle 1

Steps 1 and 2 are the same as normal.

Step 1 – Label the triangle

We need to label the sides of the triangle with H (hypotenuse), O (opposite) and A (adjacent).

right angled triangle labelled with opposite adjacent hypotenuse

Step 2 – Pick one of the SOHCAHTOA triangles

sohcahtoa

To do this, we look at the information we care given in the question and what we need to work out.

In this question, we are given the adjacent (18.6 cm) and the hypotenuse (25.1 cm). 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.

CAH triangle

Step 3 – Use the chosen triangle to solve

Now this is where things change. We are given the adjacent (A) and the hypotenuse (H), so it must be C we are trying to find.

CAH triangle covered

So we cover up the C and are left with C = A ÷ H.

Remember that C stands for cos, and that cos needs to be “of an angle”. The angle in this case is unknown so we form this equation:

cos(x) = 18.6 ÷ 25.1

Now to solve this and find x, we need to get rid of the cos. To get rid of cos, there is an inverse function on your calculator (cos-1). For each trigonometric function, there is an inverse function (you can find them in yellow on your calculator above the regular trig functions, pictured below).

inverse trigonometric functions on casio calculator

Cos-1 does the opposite of cos. Use your calculator to prove it:

Do cos of any number. Then do cos-1 of that answer. You will see that it goes back to your original number. It has the same effect as, for example, multiplying a number by 2 and then dividing it by 2. The same also works for sin / sin-1 and tan / tan-1.

So, I need to do inverse cos to both sides:

using inverse cos to find x

Example 2

Find x.

right angled triangle 2

First, label the triangle.

right angled triangle labelled with opposite adjacent hypotenuse

Now pick one of SOH, CAH or TOA.

In this question, we are given the opposite and the adjacent, so we need to use TOA.

tan opposite adjacent triangle

We are finding the angle, so we cover up the T and leave O ÷ A.

covered TOA triangle
using inverse tan to find x

Example 3

Find x.

right angled triangle 3

Label the triangle.

right angled triangle labelled with opposite adjacent hypotenuse

Now pick one of SOH, CAH or TOA.

In this question, we are given the opposite and the hypotenuse, so we need to use SOH.

sin opposite hypotenuse triangle

We are finding the angle, so we cover up the S and leave O ÷ H.

sin equals opposite divided by hypotenuse
using inverse sin to find x

So that’s how you use SOHCAHTOA to find missing angles in right-angled triangles. If you feel confident on these now and feel you can answer questions on your own, try our worksheet. If you need someone to explain this method to you in person, book a free taster session.

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