9.4PureStretch

Solving Triangle Problems

Now you have the sine rule, cosine rule and the area formula, the real skill is choosing the right one — and chaining them together for multi-step problems and bearings questions.

30 min Video by Zeeshan Zamurred Trigonometric Ratios

Edexcel AS Level Maths: 9.4 Solving Triangle ProblemsWatch the full walkthrough before the notes below.

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What you'll be able to do

Choose between the sine and cosine rules
Combine rules across several steps
Apply trigonometry to bearings and real contexts
Work to a sensible degree of accuracy

Which rule?

Pick your tool from what you are given. The cosine rule handles the cases with no ready side–angle pair; the sine rule handles the rest.

SAS or SSS \Rightarrow cosine rule

Two sides + included angle, or all three sides.

side-opposite-angle pair \Rightarrow sine rule

Whenever a matching pair is available.

Tip — No matching side–angle pair? Start with the cosine rule, which then creates one for later steps.

Multi-step problems

Harder questions need more than one rule. A typical route: use the cosine rule to get a side, then the sine rule to get an angle, then the area formula. Keep full accuracy in your working and only round at the end.

1Cosine rule for side

a

(you have SAS).

2Sine rule for a second angle, e.g.

B

3Angle sum: third angle

= 18 0^{\circ} - A - B

Answercosine rule → sine rule → angle sum

Bearings and context

Bearings are measured clockwise from north as three figures (e.g. $07 0^{\circ}$ ). Sketch the situation, mark the angles inside the triangle, then apply the rules. Always relate your final answer back to the question’s context.

Tip — Draw a clear diagram first — most marks in worded triangle questions are lost on a missing or mislabelled sketch.

Formula recap

a^{2} = b^{2} + c^{2} - 2 b c cos A

Cosine rule (SAS / SSS).

\frac{a}{sin A} = \frac{b}{sin B}

Sine rule (matching pair).

angles sum to 18 0^{\circ}

Always available as a final step.

Common mistakes to avoid

Rounding partway through a multi-step problem.

Keep full accuracy in intermediate steps; round only the final answer.

Measuring a bearing anticlockwise or from the wrong direction.

Bearings are clockwise from north, written as three figures.

Key takeaways

Cosine rule for SAS/SSS; sine rule whenever a side–angle pair exists.
Multi-step problems often chain cosine → sine → angle sum.
Bearings are three-figure angles clockwise from north — always sketch.
Keep full accuracy until the final rounding.

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