Trigonometry: SOH CAH TOA
Trigonometry helps us understand the relationship between angles and side lengths in right-angled triangles. It’s a key skill used in architecture, navigation, and engineering, and this guide will show you how to master the three core trigonometric ratios: Sine, Cosine, and Tangent.
Labelling the Triangle
The first and most important step is to correctly label the sides of a right-angled triangle in relation to a specific angle ($\theta$).
The Three Sides
- The Hypotenuse is always the longest side, opposite the right angle.
- The Opposite side is directly across from the angle $\theta$.
- The Adjacent side is next to the angle $\theta$ (but is not the hypotenuse).
The Trigonometric Ratios: SOH CAH TOA
This handy mnemonic helps you remember the three essential trigonometric ratios.
SOH
Sine = Opposite / Hypotenuse$\sin(\theta) = \frac{\text{Opp}}{\text{Hyp}}$
CAH
Cosine = Adjacent / Hypotenuse$\cos(\theta) = \frac{\text{Adj}}{\text{Hyp}}$
TOA
Tangent = Opposite / Adjacent$\tan(\theta) = \frac{\text{Opp}}{\text{Adj}}$
Worked Examples
Example 1: Finding a Missing Side
Find the length of side $x$ to 1 decimal place.- Label the sides: $x$ is the Opposite, 12cm is the Hypotenuse.
- Choose the ratio: We have O and H, so we use SOH (Sine).
- Substitute and solve: $\sin(30^\circ) = \frac{x}{12}$ $x = 12 \times \sin(30^\circ) = 6$
Example 2: Finding a Missing Angle
Find the size of angle $\theta$ to 1 decimal place.- Label the sides: 7cm is the Adjacent, 10cm is the Hypotenuse.
- Choose the ratio: We have A and H, so we use CAH (Cosine).
- Substitute: $\cos(\theta) = \frac{7}{10} = 0.7$
- Use the inverse function to find the angle: $\theta = \cos^{-1}(0.7) \approx 45.6^\circ$
Tutor Insights
🤔 Common Misunderstandings
- Mixing up Opposite and Adjacent. These sides depend entirely on which angle you’re looking from. The hypotenuse is the only one that never changes.
- Using the wrong ratio. Forgetting SOH CAH TOA or picking the wrong one for the given sides.
📝 Common Exam Mistakes
- Calculator in the wrong mode. It MUST be in Degree (DEG) mode. Radian (RAD) mode will give you the wrong answers.
- Not using the inverse function ($\sin^{-1}, \cos^{-1}, \tan^{-1}$) when finding an angle.
- Rounding too early or to the wrong number of decimal places.
Practice Questions
- Find the length of side $a$ to 1 decimal place.
- Find the size of angle $\alpha$ to 1 decimal place.
Show Answers
- Working: Sides are Opposite (a) and Hypotenuse (15). Use SOH. $\sin(35^\circ) = \frac{a}{15} \implies a = 15 \times \sin(35^\circ) \approx 8.6$ Answer: 8.6 cm.
- Working: Sides are Opposite (18) and Adjacent (12). Use TOA. $\tan(\alpha) = \frac{18}{12} = 1.5 \implies \alpha = \tan^{-1}(1.5) \approx 56.3$ Answer: 56.3°.
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