The SOH-CAH-TOA memory trick is a great way to help students understand basic trigonometry with right-angled triangles. It’s a simple way to remember how to find relationships between the sides of these triangles. Here’s what it stands for:

• SOH: Sine is the side opposite the angle divided by the longest side (hypotenuse).
• CAH: Cosine is the side next to the angle divided by the hypotenuse.
• TOA: Tangent is the side opposite the angle divided by the side next to the angle.

Understanding the Ratios

Let’s look at a right-angled triangle where angle A is one of the non-right angles. We can label the sides like this:

• Hypotenuse: The longest side, opposite the right angle.
• Opposite: The side across from angle A.
• Adjacent: The side next to angle A that is not the hypotenuse.

Examples

1. Using Sine ($\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}}$)

   • If the opposite side is 5 units and the hypotenuse is 13 units, we get: $\sin A = \frac{5}{13}$

2. Using Cosine ($\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}$)

   • If the adjacent side is 12 units, then: $\cos A = \frac{12}{13}$

3. Using Tangent ($\tan A = \frac{\text{Opposite}}{\text{Adjacent}}$)

   • With the same sides, we calculate: $\tan A = \frac{5}{12}$

Why It's Useful

This memory trick helps you remember these relationships and solve problems faster. When you have a right-angled triangle problem, thinking of SOH-CAH-TOA lets you quickly figure out which sides to use based on the angle you have. It makes it much easier to find unknown sides or angles.

In short, SOH-CAH-TOA gives you a simple and clear way to work with trigonometry, helping you tackle problems with more confidence.