SOHCAHTOA

This resource is a great reference tool for students calculating missing sides in right-angled triangles using either sine, cosine or tangent. There are three illustrated examples, one for each function. In each case, the numerator of the ratio is to be found.

SOHCAHTOA Explained: How to Solve Right-Angled Triangles with Trigonometry

The idea of SOHCAHTOA is an important concept in Trigonometry. It is specifically used for solving mainly those problems that involve right-angled triangles. If you are a student preparing for your upcoming exams or a teacher who is looking for perfect teaching resources for KS4, this worksheet will help you to master the knowledge- ‘how to use SOHCAHTOA‘ slowly and gradually.

What is SOHCAHTOA?

The idea of SOHCAHTOA can be easily defined as a mnemonic that is mostly used in trigonometry. This idea will help your students to remember the formulas for sine (sin), cosine (cos), and tangent (tan) in a right-angled triangle.

• SOH – Sine = Opposite / Hypotenuse
• CAH – Cosine = Adjacent / Hypotenuse
• TOA – Tangent = Opposite / Adjacent

This simple formula can be a huge help for your young learners to identify which trigonometric function needs to be used based on the sides and angles given in a problem.

SOHCAHTOA Triangles: Understanding the Key Terms

To correctly apply SOHCAHTOA, it is important to understand the different parts of a right-angled triangle:

1. Hypotenuse (H) – This is basically the longest side, opposite the right angle.
2. Opposite (O) – This side is directly across from the angle you are working with.
3. Adjacent (A) – This side is next to the angle (but not the hypotenuse).

Only if we can correctly identify these sides can we decide which trigonometric ratio to use.

How to Do SOHCAHTOA: Step-by-Step Guide

Step 1: Identify the Given Information

• Do you have an angle and one side?
• Are you trying to find a missing side or angle?

Step 2: Choose the Correct Formula

• If you have Opposite & Hypotenuse → You must use Sine (sin)
• If you have Adjacent & Hypotenuse → You need to use Cosine (cos)
• If you have Opposite & Adjacent → The students must use Tangent (tan)

Step 3: Solve for the Missing Value

• Finding a Side: For this purpose, we need to rearrange the formula to solve for the unknown length.
• Finding an Angle: In this case, we need to use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹) on your calculator.

Examples of Using SOHCAHTOA Triangles

Example 1: Finding a Missing Side with Sine

Problem: Find the length of the opposite side in a triangle where the hypotenuse is 7 cm, and the angle is 30°.

Solution:

\(
\sin 30^\circ = \frac{\text{Opposite}}{\text{Hypotenuse}} \\
\sin 30^\circ = \frac{x}{7} \\
x = 7 \times \sin 30^\circ \\
x = 7 \times 0.5 \\
x = 3.5 \text{ cm}
\)

Example 2: Finding an Angle with Cosine

Problem: A triangle has an adjacent side of 4 cm and a hypotenuse of 8 cm. Find the angle.

Solution:

\(
\cos \theta = \frac{4}{8} \\
\cos \theta = 0.5 \\
\theta = \cos^{-1}(0.5) \\
\theta = 60^\circ
\)

Example 3: Finding a Side with Tangent

Problem: A right-angled triangle has an adjacent side of 9 cm, and the angle is 50°. Find the opposite side.

Solution:

\(
\tan 50^\circ = \frac{\text{Opposite}}{9} \\
\text{Opposite} = 9 \times \tan 50^\circ \\
\text{Opposite} = 9 \times 1.1918 \\
\text{Opposite} = 10.7 \text{ cm}
\)

Common Mistakes When Using SOHCAHTOA

We need to be really careful while using SOHCAHTOA. There can be some common mistakes that might be an issue while using SOHCAHTOA, like-

• Choosing the wrong sides – Students need to double-check which sides are opposite, adjacent, and hypotenuse.
• Forgetting to use inverse functions – While finding angles, they must use sin⁻¹, cos⁻¹, tan⁻¹ on their calculator.
• Using degrees instead of radians – The students must make sure that their calculator is in the correct mode (DEG mode for most school problems).

Why is SOHCAHTOA Important?

1. Essential for GCSE and High School Maths – SOHCAHTOA is a core and important topic in trigonometry.
2. Used in Real-Life Applications – From engineering to navigation, the basic idea of this concept helps solve practical problems.
3. Forms the Basis for Advanced Trigonometry – Understanding these ratios is really important before moving on to the sine and cosine rules of Trigonometry.