Advanced Trigonometry
Beyond right-angled triangles, trigonometry is a powerful tool for solving problems in 3D shapes and any triangle. This guide will walk you through the key higher-tier skills of 3D trigonometry, the Sine Rule, and the Cosine Rule.
The Advanced Trigonometry Toolkit
Trigonometry in 3D
The key to solving 3D problems is to find the hidden 2D right-angled triangles within the shape. You often need to use Pythagoras’ theorem on one face to find a length, which then becomes a side in another right-angled triangle inside the shape.
Non-Right-Angled Triangles
For any triangle that doesn’t have a 90° angle, SOH CAH TOA doesn’t work. Instead we use the Sine Rule or Cosine Rule. The standard labelling convention is shown below.
Choosing the Right Tool
Knowing which rule to use is the most important skill for non-right-angled triangles.
Use the Sine Rule when…
You have a matching pair of a side and its opposite angle, plus one other piece of information.
Use the Cosine Rule when…
You have two sides and the included angle (SAS), or you have all three sides (SSS).
Use the Area Rule when…
You have two sides and the included angle (SAS) and need to find the area.
Worked Examples
Example 1: Sine Rule
Find the length of side $x$.
- Find the third angle: $180° – 50° – 65° = 65°$.
- We have a pair: side 12 cm is opposite 65°.
- Sine Rule: $\frac{x}{\sin 50°} = \frac{12}{\sin 65°}$.
- Rearrange: $x = \frac{12 \times \sin 50°}{\sin 65°} \approx 10.1$.
Answer: 10.1 cm (to 1 d.p.)
Example 2: Cosine Rule
Find the length of side $y$.
- Two sides and the included angle (SAS) → Cosine Rule.
- $y^2 = 8^2 + 10^2 – 2(8)(10)\cos 70°$.
- $y^2 = 64 + 100 – 160 \times 0.342… = 109.27…$
- $y = \sqrt{109.27…} \approx 10.5$.
Answer: 10.5 cm (to 3 s.f.)
Tutor Insights
🤔 Common Misunderstandings
- Not seeing the hidden 2D triangles in 3D problems. The key is to sketch them separately.
- Mixing up the Sine and Cosine Rules. Use this quick check: do you have a matching side/angle pair? Yes → Sine Rule. No → Cosine Rule.
- Incorrectly identifying the included angle for the Cosine Rule or Area formula.
📝 Common Exam Mistakes
- Calculator in wrong mode. It must be in Degree (DEG) mode.
- Algebraic errors when rearranging the Cosine Rule to find an angle.
- Rounding intermediate steps too early, leading to an inaccurate final answer.
- Using SOH CAH TOA on non-right-angled triangles.
Practice Questions
- In $\triangle LMN$, $LM = 15$ cm, $\angle L = 35°$, and $\angle N = 68°$. Find the length of $MN$ to 3 s.f.
- A triangular garden has sides of 12 m, 18 m, and 25 m. Find the largest angle to 1 d.p.
- Calculate the area of $\triangle XYZ$ where $XY = 10$ cm, $XZ = 14$ cm, and $\angle X = 110°$. Give your answer to 3 s.f.
Show Answers
- Working: $\angle M = 180° – 35° – 68° = 77°$. Sine Rule: $\frac{MN}{\sin 35°} = \frac{15}{\sin 68°} \implies MN = \frac{15\sin 35°}{\sin 68°} \approx 9.27$.
Answer: 9.27 cm. - Working: Largest angle is opposite the 25 m side. Cosine Rule: $\cos A = \frac{12^2 + 18^2 – 25^2}{2 \times 12 \times 18} = \frac{-157}{432} \implies A = \cos^{-1}\!\left(\frac{-157}{432}\right) \approx 111.3°$.
Answer: 111.3°. - Working: Area $= \frac{1}{2}(10)(14)\sin 110° \approx 65.8$.
Answer: 65.8 cm².
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