📐 SOH CAH TOA — Right-Angled Triangle Trigonometry (NSW Year 9–10)
📏 Law of Sines (Sine Rule) — Non-Right-Angled Triangles (NSW Year 10)
🧠 Trigonometry Test — Mixed (SOH CAH TOA + Sine Rule + Elevation/Depression)
Trigonometry Made Simple: SOH CAH TOA + Sine Rule
In NSW Year 9 and 10, trigonometry is mainly about solving triangles. You’ll use SOH CAH TOA for right-angled triangles, and the Sine Rule for non-right-angled triangles (Year 10). This guide stays clear, concise, and exam-relevant.
Looking for more support? Visit our free maths worksheets and practice tests pages.
1) Right-Angled Triangles (SOH CAH TOA)
In a right-angled triangle, pick the ratio that matches the sides you have and the side you need.
SOH: \(\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}\)
CAH: \(\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}\)
TOA: \(\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}\)
Tip: “Opposite” and “Adjacent” depend on which angle \(\theta\) you are using.
Quick refresh on triangle parts? See: Geometry basics.
| What you know | Use |
|---|---|
| Opposite + Hypotenuse | Sine \(\sin\theta\) |
| Adjacent + Hypotenuse | Cosine \(\cos\theta\) |
| Opposite + Adjacent | Tangent \(\tan\theta\) |
Example: Find the opposite side
A right-angled triangle has an angle of \(30^\circ\) and a hypotenuse of 10 cm. Find the opposite side.
- Opposite and hypotenuse are involved → use sine.
- \(\sin 30^\circ=\frac{\text{Opposite}}{10}\)
- \(\frac{1}{2}=\frac{\text{Opposite}}{10}\Rightarrow \text{Opposite}=5\)
Example: Find the angle \(\theta\)
In a right-angled triangle, the opposite side is 6 m and the adjacent side is 8 m. Find \(\theta\).
- Opposite and adjacent are involved → use tangent.
- \(\tan\theta=\frac{6}{8}=0.75\)
- \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\)
Want calculator help? See: How to use a scientific calculator.
2) Angles of Elevation and Depression
- Angle of elevation: looking up from the horizontal.
- Angle of depression: looking down from the horizontal.
These questions form a right-angled triangle — SOH CAH TOA applies.
More real-life triangles: Measurement & applications.
Example: Ladder against a wall
A ladder is 8 m long and makes an angle of \(60^\circ\) with the ground. How high does it reach?
- Height is opposite; ladder is hypotenuse → use sine.
- \(\sin 60^\circ=\frac{\text{Height}}{8}\Rightarrow \text{Height}=8\sin 60^\circ\)
- \(\text{Height}=8\times\frac{\sqrt3}{2}=4\sqrt3\approx 6.93\ \text{m}\)
3) The Law of Sines (Sine Rule) – Non-Right-Angled Triangles
If the triangle is not right-angled, use the Sine Rule (a key NSW Year 10 topic).
Sine Rule: \(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
Use it when:
- You know two angles and one side (AAS / ASA), or
- You know two sides and a non-included angle (SSA).
External reference (good for extra reading): Sine Rule explained (Math is Fun).
Example: Find a side in a non-right triangle
In triangle \(ABC\): \(A=40^\circ\), \(B=60^\circ\), and side \(a=10\ \text{cm}\). Find side \(b\).
- Match each side with its opposite angle: \(a\leftrightarrow A\), \(b\leftrightarrow B\).
- \(\frac{a}{\sin A}=\frac{b}{\sin B}\Rightarrow \frac{10}{\sin 40^\circ}=\frac{b}{\sin 60^\circ}\)
- \(b=\frac{10\sin 60^\circ}{\sin 40^\circ}\approx \frac{10\times 0.8660}{0.6428}\approx 13.47\ \text{cm}\)
Round at the end to avoid accuracy errors.
4) Common Mistakes (Avoid Easy Marks Lost)
- Mixing up opposite vs adjacent (depends on \(\theta\)).
- Using the wrong ratio (sine/cosine/tangent).
- Forgetting inverse trig when finding an angle.
- Calculator not in DEG.
Related internal reading: Angles & triangles basics.
- Not matching the side with its opposite angle.
- Substituting values into the wrong fraction.
- Rounding too early.
- Forgetting to label units (cm, m) and degrees \((^\circ)\).
Extra support: Year 10 Maths NSW syllabus guide.
For NSW Year 9–10, trigonometry is about being confident with SOH CAH TOA in right-angled triangles, then extending to non-right triangles using the Sine Rule in Year 10. If you take problems step-by-step and choose the correct ratio, trig becomes very manageable.
Please note: School programs may vary slightly, so always check your teacher’s scope and sequence.
Need help fast? Book a session here: Contact Aussie Math Tutor NSW.
Useful references for students and parents (internal + external):
