Unknown side in the numerator
Identify the two sides, choose their ratio and multiply by the known denominator
- Choose from the two sides Use the marked angle to identify the given side and wanted side. Choose the ratio that contains both
- Multiply both sides When the wanted length is the numerator, multiply both sides of the equation by the known denominator \(\displaystyle \begin{aligned}\frac{x}{k}&=r\\k\times\frac{x}{k}&=kr\\x&=kr\end{aligned}\)
- Round the final length Use degree mode and keep the trigonometric value in the calculator until the length is found
Worked example
Find the adjacent length in centimetres, to one decimal place
\[\begin{gathered}\theta=46^{\circ}\\\text{hypotenuse}=21\text{ cm}\\\text{adjacent}=x\text{ cm}\end{gathered}\]
- \(\displaystyle \cos46^{\circ}=\frac{x}{21}\)Cosine links the adjacent side and hypotenuse
- \(\displaystyle 21\cos46^{\circ}=21\times\frac{x}{21}\)Multiply both sides by the known denominator
- \(\displaystyle x=21\cos46^{\circ}\)Cancel the denominator and write the unknown first
- \(\displaystyle x=14.5878257796\ldots\)Evaluate in degree mode and keep the calculator value
- \(\displaystyle x\approx14.6\text{ cm}\)Round once to one decimal place. The result is shorter than the known hypotenuse
Common mistakes
Dividing for an unknown numerator
✕\(\displaystyle \begin{aligned}\sin58^{\circ}&=\frac{x}{17}\\x&=\frac{17}{\sin58^{\circ}}\end{aligned}\)✓\(\displaystyle \begin{aligned}\sin58^{\circ}&=\frac{x}{17}\\x&=17\sin58^{\circ}\end{aligned}\)
Multiplication by the denominator undoes the division in the original ratioDifficulty levels
introFind a numerator side using sine, cosine or tangent from a labelled triangle with a whole-number given length
standardFind a numerator side using sine, cosine or tangent with decimal lengths and different triangle orientations
challengeFind a full baseline by adding two numerator lengths from right triangles that share a perpendicular altitude. Round only the final sum
examChoose the numerator length from three values using sine, cosine or tangent
Questions about this topic
Why do I multiply?+
The unknown length is divided by a known length. Multiplying both sides by that known length isolates the unknown
Must the answer be smaller than the known side?+
It must be smaller when the known side is the hypotenuse. With tangent, the two shorter sides can have either order
When should I round?+
Round the requested final length. Keep the trigonometric value at full calculator precision while calculating
Can I use a calculator?+
A calculator is allowed unless a question is marked no 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