Trigonometric Ratios, Special Angles & Problem Solving
Master SOH CAH TOA, the special angle values, reciprocal ratios, and solving right-angled triangle problems — with full worked examples.
Trigonometry
Table of Contents
Trigonometry: The Logic of Right-Angled Triangles#
Trigonometry connects angles to side lengths. In Grade 10, everything happens inside the right-angled triangle. If you know one angle and one side, you can find everything else.
The Right-Angled Triangle & Labelling#
The key to getting trig right is labelling from the correct angle. The sides change names depending on which angle you’re working from:
- Hypotenuse: Always the longest side, always opposite the right angle ($90°$).
- Opposite: The side across from the angle $\theta$ you’re working with.
- Adjacent: The side next to the angle $\theta$ (that isn’t the hypotenuse).
⚠️ If the angle changes, Opposite and Adjacent swap. Always re-label when you switch angles!
SOH CAH TOA#
The three ratios that connect angles to sides:
| Ratio | Formula | Memory aid |
|---|---|---|
| $\sin\theta$ | $\frac{\text{Opposite}}{\text{Hypotenuse}}$ | Sine = Opp / Hyp |
| $\cos\theta$ | $\frac{\text{Adjacent}}{\text{Hypotenuse}}$ | Cosine = Adj / Hyp |
| $\tan\theta$ | $\frac{\text{Opposite}}{\text{Adjacent}}$ | Tangent = Opp / Adj |
Special Angles You Must Memorise#
| $\theta$ | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ |
|---|---|---|---|
| $30°$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| $45°$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $60°$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
💡 Notice: $\sin 30° = \cos 60°$ and $\sin 60° = \cos 30°$. This is the co-function relationship — it becomes very important in Grade 11.
Deep Dive#
- Trig Ratios, Special Angles & Problem Solving — full worked examples for finding sides, finding angles, reciprocal ratios, elevation & depression problems
🚨 Common Mistakes#
- Labelling from the wrong angle: Opposite and Adjacent swap when you change angles. ALWAYS re-label.
- Calculator in wrong mode: Must be in DEG (degrees), not RAD.
- Inverse trig confusion: $\sin^{-1}$ is NOT $\frac{1}{\sin}$. It means “what angle has this sine value?”
- Forgetting the right angle: SOH CAH TOA only works in right-angled triangles.
🔗 Related Grade 10 topics:
- Analytical Geometry — gradient = $\tan\theta$ and Pythagoras gives the distance formula
📌 Where this leads in Grade 11: Trigonometry — Beyond Right Angles — CAST diagram, reduction formulas, identities, and solving any triangle
⏮️ Probability | 🏠 Back to Grade 10 | ⏭️ Euclidean Geometry