# SOHCAHTOA (Trigonometry) Welcome to Trigonometry! At Grade 5, you will learn how to connect the angles inside a right-angled triangle to the lengths of its sides. **What you'll learn:** - How to correctly label the sides of a right-angled triangle. - What the acronym "SOH CAH TOA" means and how to choose the right formula. - How to calculate a missing side length. - How to calculate a missing angle using inverse trigonometric functions. ## 1. Labelling the Triangle Before doing any calculations, you must label your triangle correctly. The names of the sides change depending on which angle you are looking at. Let's call our angle $\theta$ (the Greek letter "theta"). ![Right-angled triangle labelled with Hypotenuse, Opposite, and Adjacent](https://assets.yesgenie.com/notes/8b8563ab-4e83-4c1e-8f44-cbceb72bca1c.png) - **Hypotenuse (H):** Always the longest side, and always directly opposite the right angle. - **Opposite (O):** The side directly opposite the angle you are working with ($\theta$). - **Adjacent (A):** The side next to the angle $\theta$. It connects the right angle and the angle $\theta$. ## 2. The Formulas There are three main trigonometric ratios: **sine**, **cosine**, and **tangent** (shortened to sin, cos, and tan on your calculator). We use the memory trick **SOH CAH TOA** to remember the formulas. - **SOH:** $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - **CAH:** $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - **TOA:** $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ If you struggle to rearrange formulas, you can put SOH, CAH, and TOA into formula triangles, just like you would for Speed, Distance, Time. Put the middle letter (O, A, or O) at the top of the triangle. Cover up what you want to find! ## 3. Finding a Missing Side To find a missing side length, follow a simple 4-step process: label the sides, pick your formula, substitute the numbers, and solve. A right-angled triangle has an angle of 35°. The hypotenuse is 20 cm. Find the length of the side opposite the 35° angle, labelled $x$. 1. Label the sides. We want the **Opposite** (which is $x$), and we know the **Hypotenuse** (which is 20). 2. Choose the formula. We have O and H, which means we need **SOH**. 3. Write out the formula and substitute the values: $$ \sin(35^\circ) = \frac{x}{20} $$ 4. Solve for $x$ by multiplying both sides by 20. Type this into your calculator to get the final answer: $$ \begin{aligned} x &= 20 \times \sin(35^\circ) \\ x &= 11.47\text{ cm (to 2 d.p.)} \end{aligned} $$ ### When the unknown is on the bottom Sometimes, the side you are trying to find ends up on the bottom (the denominator) of the fraction. This just adds one extra step to your solving process. A right-angled triangle has an angle of 42°. The side opposite the angle is 11 cm. Find the length of the adjacent side, $x$. 1. Label the sides. We know the **Opposite** (11) and we want the **Adjacent** ($x$). 2. Choose the formula. O and A means we use **TOA**. 3. Substitute the values: $$ \tan(42^\circ) = \frac{11}{x} $$ 4. To solve, first multiply by $x$ to get it off the bottom, then divide by $\tan(42^\circ)$. Essentially, $x$ and $\tan(42^\circ)$ swap places: $$ \begin{aligned} x \times \tan(42^\circ) &= 11 \\ x &= \frac{11}{\tan(42^\circ)} \\ x &= 12.22\text{ cm (to 2 d.p.)} \end{aligned} $$ ## 4. Finding a Missing Angle If you know two side lengths, you can find a missing angle. You still use SOH CAH TOA, but at the end, you need to use the **inverse** trigonometric functions ($\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) on your calculator. You usually get these by pressing the `SHIFT` or `2ND` button before the sin, cos, or tan button.