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Solve a practical triangle problem end to end

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Which method is the right first step for this route?

\[AB=24\ \mathrm{km},\quad BC=24\ \mathrm{km},\quad \angle ABC=64^{\circ}\]
Year 11, VCE Units 1 & 2, Chapter 11, 3 difficulty levels, Calculator allowedBuild a worksheet

Chaining triangle calculations

A practical problem may require more than one triangle calculation. Choose the rule from the given data, keep the first result unrounded, and then use it in the next relation

\[d^{2}=a^{2}+b^{2}-2ab\cos C,\qquad t=\frac{60d}{v}\]
  1. Find the working angle For a journey triangle, use the angle between the two legs, not a bearing measured from north \(\displaystyle 35^{\circ}+180^{\circ}=215^{\circ},\quad215^{\circ}-151^{\circ}=64^{\circ}\)
  2. Choose the rule from the data Two sides and their included angle use the cosine rule \(\displaystyle d^{2}=24^{2}+24^{2}-2(24)(24)\cos64^{\circ}\)
  3. Keep the first result unrounded Carry the full first-stage value into the next calculation and round only the final quantity being reported \(\displaystyle d=\sqrt{24^{2}+24^{2}-2(24)(24)\cos64^{\circ}}\approx25.4361\ \mathrm{km}\)
  4. Convert distance to time When distance is in kilometres and speed is in kilometres per hour, multiply distance divided by speed by 60 for minutes \(\displaystyle t=\frac{60d}{12}\approx127.2\ \mathrm{min}\)

Worked example

A field survey team measures a baseline AB of 52 m. The angles at A and B to an inaccessible point C are 41° and 63°. Find AC, then find the area of triangle ABC, giving both answers to 1 decimal place

\[\triangle ABC:\quad AB=52\ \mathrm{m},\quad \angle A=41^{\circ},\quad \angle B=63^{\circ}\]
  1. \(\displaystyle C=180^{\circ}-41^{\circ}-63^{\circ}=76^{\circ}\)The third angle is needed because side AB is paired with angle C in the sine rule
  2. \(\displaystyle \frac{AC}{\sin63^{\circ}}=\frac{52}{\sin76^{\circ}}\)The baseline AB is opposite angle C, and side AC is opposite angle B
  3. \(\displaystyle AC=52\frac{\sin63^{\circ}}{\sin76^{\circ}}\approx47.75074\ \mathrm{m}\approx47.8\ \mathrm{m}\)Solving the sine-rule equation gives an unrounded sight line of about 47.75074 m
  4. \(\displaystyle A_{\triangle ABC}=\frac{1}{2}(52)(47.75074)\sin41^{\circ}\approx814.5\ \mathrm{m}^{2}\)Use the full first-stage value in the area formula and round the final area to 1 decimal place

Common mistakes

Choosing a rule from the data
✕\(\displaystyle d=\sqrt{24^{2}+24^{2}}=\sqrt{1152}\approx33.9\ \mathrm{km}\)✓\(\displaystyle d=\sqrt{24^{2}+24^{2}-2(24)(24)\cos64^{\circ}}\approx25.4\ \mathrm{km}\)
Two side lengths alone do not make a right triangle; the included angle is 64°, so the cosine rule is required
Using a raw bearing as the angle
✕\(\displaystyle d=\sqrt{24^{2}+24^{2}-2(24)(24)\cos35^{\circ}}\approx14.4\ \mathrm{km}\)✓\(\displaystyle d=\sqrt{24^{2}+24^{2}-2(24)(24)\cos64^{\circ}}\approx25.4\ \mathrm{km}\)
The cosine rule needs the angle between the two legs at B; the raw first bearing is measured from north, not between the legs
Rounding the intermediate distance
✕\(\displaystyle d=25.4\ \mathrm{km},\quad t=\frac{60(25.4)}{12}=127\ \mathrm{min}\)✓\(\displaystyle d=25.4361\ldots\ \mathrm{km},\quad t=\frac{60d}{12}\approx127.2\ \mathrm{min}\)
Rounding the first distance before the time calculation changes the final value, so carry the full distance and round only the time

Difficulty levels

standardSolve a three-part practical chain by choosing a rule, finding a first length, and completing a second calculation
challengeSolve a multi-stage parcel, right-triangle or solid-frame problem and interpret the final quantity
examChoose the final value from options containing intermediate-rounding, geometry or angle errors

Questions about this topic

Why is the working angle 64° rather than 35°?+

The 35° value is a bearing from north. The cosine rule needs the angle between the two route legs, so use the back bearing 215° and subtract the second bearing 151° to get 64°

Why should the first distance not be rounded before the next stage?+

The second calculation uses the first distance as an input. Early rounding changes that input and can change the final answer, so round only the quantity requested at the end

Why is distance multiplied by 60 when finding time in minutes?+

Dividing distance in kilometres by speed in kilometres per hour gives hours. Multiplying by 60 changes hours into minutes

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 11: Trigonometry

In this chapter

Name sides and state trigonometric ratios11A Trigonometric ratios: Evaluate a trigonometric ratio, and find an angle from its valueChoose a trigonometric ratioFind an unknown side of a right-angled triangle11C Finding unknown angles: Evaluate a trigonometric ratio, and find an angle from its valueFind an angle of a right-angled triangle from two sidesSolve a practical right-angled-triangle problemAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsUse the sine rule in a non-right-angled triangleThe ambiguous case of the sine ruleUse the cosine rule in a non-right-angled triangle11H Cosine rule: Choose the rule a triangle problem needs11H Cosine rule: Solve a practical triangle problem end to end11I Triangle area: Choose the rule a triangle problem needsFind the area of a triangle11I Triangle area: Solve a practical triangle problem end to end