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Choose a trigonometric ratio

Start a set of 8
Step 3 of 10

Choose first, rearrange second

An unknown denominator does not automatically mean sine. Its side pair determines the function; its position determines the rearrangement

  1. 1Use the marked angle Name the given and wanted sides relative to that angle. The hypotenuse is always opposite the right angle
  2. 2Choose 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. 3Recheck after an angle change The two shorter sides exchange roles when the marked acute angle changes. Check the pair again
Year 10, Chapter 9, 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

The known opposite side is 7 cm and the wanted adjacent side is x. Choose the ratio and isolate x

\[\tan\theta=\frac{7}{x}\]
  1. \(\displaystyle x\tan\theta=7\)Multiply both sides by x
  2. \(\displaystyle x=\frac{7}{\tan\theta}\)Divide by the nonzero tangent value

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

standardChoose the ratio connecting a known numerator side and a wanted denominator 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 10 Mathematics, chapter 9: Pythagoras and trigonometry

In this chapter

9A Finding the hypotenuse: Find a side of a right-angled triangleDetermine whether three sides form a right-angled triangle9B Finding a shorter side: Find a side of a right-angled triangle9C Pythagoras' theorem in 2D and 3D: Find a side of a right-angled trianglePythagoras in three dimensionsName sides and state trigonometric ratios9D Trigonometric ratios: Evaluate a trigonometric ratio9D Trigonometric ratios: Choose a trigonometric ratio9E Unknown side in the numerator: Find an unknown side of a right-angled triangle9E Unknown side in the numerator: Solve a practical right-angled-triangle problem9E Unknown side in the numerator: Evaluate a trigonometric ratio9E Unknown side in the numerator: Choose a trigonometric ratio9F Unknown side in the denominator: Find an unknown side of a right-angled triangle9F Unknown side in the denominator: Solve a practical right-angled-triangle problem9F Unknown side in the denominator: Choose a trigonometric ratioFind an angle of a right-angled triangle from two sides9G Finding angles with trigonometry: Solve a practical right-angled-triangle problem9G Finding angles with trigonometry: Evaluate a trigonometric ratio, and find an angle from its valueAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsTrigonometry in three dimensions