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.
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).
Step 2 – Pick one of the SOHCAHTOA triangles
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.
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.
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).
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:
Example 2
Find x.
First, label the triangle.
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.
We are finding the angle, so we cover up the T and leave O ÷ A.
Example 3
Find x.
Label the triangle.
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.
We are finding the angle, so we cover up the S and leave O ÷ H.
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.