# Trigonometry: TOA CAH SOH

Updated: Jun 22, 2021

Today, Math Lobby will be discussing about three of the most important trigonometry functions that you will have to know for sure as an O/N-Level Math student. These functions will be very useful in solving many trigonometry-related questions involving a right-angled triangle and also in topics such as ** pythagoras theorem**, real world problems,

**and**

__congruency__**and many other topics in the Secondary Math**

__similarity__**syllabus.**

Furthermore, it tested yearly in the **O Level E Math**, **O Level A Math**, **N Level Math** and **N Level A Math** examination. It will be taught to **Secondary 2 and 3 Math** students. Without further ado, let us learn how to solve trigonometry questions using TOA CAH SOH today!

__In this note, you will learn:__

__In this note, you will learn:__

### · __What is “TOA CAH SOH”?__

__What is “TOA CAH SOH”?__

### · __What can we use “TOA CAH SOH” to solve?__

__What can we use “TOA CAH SOH” to solve?__

First of all, let’s take a look at the right-angled triangle.

You will notice that for the different sides of the triangle, there is a name for it. How do we know which name is for which side of the triangle? There are a few characteristics to each side of the triangle:

The side which is opposite to the right angle or is the longest out of the whole triangle is called the **hypotenuse**.

The side which sits at the base of the hypotenuse and is touching the right angle is called the **adjacent**.

The side which is touching the right angle, but is opposite of the hypotenuse is called the **opposite**.

After knowing the names of the sides of the triangle, next we will be introducing the three trigonometric functions, namely “**TOA**”, “**CAH**” and “**SOH**”. This might seem unfamiliar but we will break it down and see what it stands for:

“**TOA**” stands for **T**angent(θ) = **O**pposite / **A**djacent

“**CAH**” stands for **C**osine(θ) = **A**djacent / **H**ypotenuse

And lastly, “**SOH**” stands for **S**ine(θ) = **O**pposite / **H**ypotenuse

A simple abbreviation to help you memorize: **T**rigonometry’s **O**rigin **A**nd **C**ontinuous **A**pplication **H**elp **S**ociety **O**perate **H**armoniously

Click on the following link to check on our article (__ How to use Mnemonics to easily memorise content__)

__How can we use “TOA CAH SOH” to solve Secondary Math questions?__

__How can we use “TOA CAH SOH” to solve Secondary Math questions?__

Using “TOA CAH SOH”, we can solve for:

1) The length of any sides

2) The angle in between any of the sides

Let’s take a look at how it works:

Given that we have a right-angled triangle with a side of **5cm**, and an angle of **60°**. Find **x**.

First of all, let’s identify what we are given:

· **x**, which is the hypotenuse since it is the longest side and is opposite the right angle

· A side of 5cm which fulfils the criteria of 1) being at the base of the hypotenuse and 2) it is touching the right angle; hence it is the **adjacent**.

· An angle of 60°, which is in between the hypotenuse and the adjacent.

**Math Lobby**__ Solution__:

We can use the trigonometric function “**CAH**” to solve this question, because we know the length of the **adjacent** side and the angle in between the hypotenuse and the adjacent, which allows us to solve for the length of the hypotenuse. Let’s see how it’s done:

Cos(θ) = Adjacent / Hypotenuse

Let’s substitute in all the values and it will give us: Cos (**60°**) = **5 **/ **x**

0.5 = 5 / **x**

0.5**x** = 5

Therefore, **x** = **10cm**

And that is how you solve for the side of a right-angled triangle using “TOH CAH SOH”!

Now, let’s take a look at another example:

Given that we have a right-angled triangle with sides of 5cm, 6cm and 8cm. Find **x**.

Now this question wants us to find an angle in the triangle instead of the length of a side of the triangle. Let’s take a look at how it’s done:

First of all, let’s identify what we are given:

· The sides of the triangles with the length of 8cm, 6cm and 5cm. We know that since the hypotenuse is the longest side of the triangle, the side with 8cm must be the **hypotenuse**. Hence, the side that is opposite of the hypotenuse will be 6cm and the side at the base of the hypotenuse will be 5cm.

· **x**, which is the angle between the hypotenuse and the opposite.

__Math Lobby____ Solution:__

We can use the trigonometric function “**SOH**” to solve for the angle, **x **because this trigonometric function involves both the **hypotenuse** and the **opposite**, which are both known values to us! Let’s see how it’s done:

Sin(θ) = Opposite / Hypotenuse

Sin(**x**) = **6** / **8**

Therefore, **x** =sin^-1 (**6 **/ **8**)

= **48.59° **

~ **48.6° (1 d.p)**

***Note: There will be another answer when solving for the arc sin-1 6 / 8, but we will omit that because in a right-angled triangle, the sum of the two other angles cannot be >90°.**

And hence, that is how you solve for an unknown angle in a right-angled triangle using “TOA CAH SOH”!

That’s all we have for today, and __ Math Lobby__ hope that all of you have a good understanding of what is “TOA CAH SOH” and how the application of “TOA CAH SOH” can help us to solve Secondary Math questions related to a right-angled triangle. And as always: Work hard, stay motivated and we wish all students a successful and enjoyable journey with

**Math Lobby!**

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