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Choose a trigonometric ratio

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A schematic right triangle has a right angle at the lower right and theta at the lower left. The sloping side is labelled nineteen centimetres and the vertical side x centimetres. Not to scalex cm19 cmθ

Which ratio links the labelled sides? The diagram is not to scale

Year 11, VCE Units 1 & 2, Chapter 11, 4 difficulty levels, No calculatorBuild a worksheet

Selecting a ratio

Match the two involved side roles to a trigonometric ratio

  1. Use the marked angle Name the given and wanted sides relative to that angle. The hypotenuse is always opposite the right angle
  2. Choose the matching pair Choose the ratio that contains both involved sides \(\displaystyle \begin{aligned}\sin\theta&=\frac{\text{opposite}}{\text{hypotenuse}}\\\cos\theta&=\frac{\text{adjacent}}{\text{hypotenuse}}\\\tan\theta&=\frac{\text{opposite}}{\text{adjacent}}\end{aligned}\)
  3. Recheck after an angle change The two shorter sides exchange roles when the marked acute angle changes. Check the pair again

Worked example

Choose a ratio for the stated given and wanted sides of a right triangle

\[\begin{gathered}\text{given side: adjacent to }\theta\\\text{wanted side: opposite }\theta\end{gathered}\]
  1. \(\displaystyle \begin{gathered}\text{involved pair:}\\\text{opposite and adjacent}\end{gathered}\)Both shorter sides are involved. The hypotenuse is not needed
  2. \(\displaystyle \tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)Tangent contains exactly that pair, so choose tangent

Common mistakes

Treating the wanted side as always opposite
✕\(\displaystyle \sin\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)✓\(\displaystyle \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
A side is named relative to the marked angle. A wanted adjacent side and a known hypotenuse need cosine

Difficulty levels

introChoose sine, cosine or tangent from a labelled triangle with a known side and a wanted numerator side

Questions about this topic

Does the unknown letter determine the ratio?+

No. Name both involved sides relative to the marked angle, then choose the ratio containing them

What changes when the other acute angle is marked?+

The shorter sides exchange opposite and adjacent roles. The hypotenuse stays the same

When do I use tangent?+

Use tangent when the known and wanted sides are the two shorter sides

Can I use a calculator?+

No. This topic is practised without a 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