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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 9, Chapter 3, 1 difficulty level, 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 9 Mathematics, chapter 3: Pythagoras and trigonometry

In this chapter

3A Finding the hypotenuse: Find a side of a right-angled triangle3B Finding a shorter side: Find a side of a right-angled triangleApply Pythagoras' theorem in two-dimensional contextsPythagoras in three dimensions (optional extension)Name sides and state trigonometric ratiosChoose a trigonometric ratio3F Unknown side in the numerator: Evaluate a trigonometric ratio3F Unknown side in the numerator: Find an unknown side of a right-angled triangle3G Unknown side in the denominator: Find a side in the denominator3H Finding angles with trigonometry: Find an angle from a trigonometric ratioFind an angle of a right-angled triangle from two sidesAngles of elevation and depressionSolve a practical right-angled-triangle problem (optional practice)Read and state three-figure bearingsSolve a navigation problem with bearings