Find a side in the denominator
Use the two involved side roles to select sine, cosine or tangent. Clear an unknown denominator before isolating the side length
\[\begin{aligned}\sin\theta&=\frac{o}{h}\\\cos\theta&=\frac{a}{h}\\\tan\theta&=\frac{o}{a}\end{aligned}\]
- Clear the denominator The letters stand for a positive ratio value, a known length and an unknown length. Multiply both sides by the unknown length \(\displaystyle \begin{aligned}r&=\frac{k}{x}\\xr&=x\times\frac{k}{x}\\xr&=k\end{aligned}\)
- Isolate the length Divide both sides by the same nonzero ratio value \(\displaystyle \begin{aligned}\frac{xr}{r}&=\frac{k}{r}\\x&=\frac{k}{r}\end{aligned}\)
- Keep calculator precision Use degree mode. Enter the full trigonometric calculation and round only the final length to the requested decimal places
- Check the side roles An unknown hypotenuse must exceed the known leg. An unknown adjacent leg can be longer or shorter than the opposite leg, depending on the marked angle
Worked example
In a right-angled triangle, the given leg is adjacent to the marked angle and the unknown side is the hypotenuse. Calculate the unknown length to two decimal places
\[\begin{aligned}\theta&=26^{\circ}\\a&=18.7\text{ cm}\\h&=x\text{ cm}\end{aligned}\]
- \(\displaystyle \cos26^{\circ}=\frac{18.7}{x}\)Cosine links the adjacent leg and the hypotenuse
- \(\displaystyle x\cos26^{\circ}=x\times\frac{18.7}{x}\)Multiply both sides by the unknown length
- \(\displaystyle x\cos26^{\circ}=18.7\)Cancel the common factor on the right
- \(\displaystyle \frac{x\cos26^{\circ}}{\cos26^{\circ}}=\frac{18.7}{\cos26^{\circ}}\)Divide both sides by the cosine value
- \(\displaystyle x=\frac{18.7}{\cos26^{\circ}}\)The unknown is now isolated
- \(\displaystyle x\approx20.805656\text{ cm}\)Evaluate in degree mode. The displayed decimal is shortened; retain the full calculator value
- \(\displaystyle x\approx20.81\text{ cm}\)Round to two decimal places. The hypotenuse is longer than the known leg
Common mistakes
Using the numerator method
✕\(\displaystyle x=17.3\sin41^{\circ}\)✓\(\displaystyle \begin{aligned}\sin41^{\circ}&=\frac{17.3}{x}\\x\sin41^{\circ}&=17.3\\x&=\frac{17.3}{\sin41^{\circ}}\end{aligned}\)
With an unknown denominator, first multiply both sides by the unknown. Then divide by the sine valueRounding the divisor early
✕\(\displaystyle \begin{aligned}x&\approx\frac{18.7}{0.90}\\x&\approx20.78\text{ cm}\end{aligned}\)✓\(\displaystyle \begin{aligned}x&=\frac{18.7}{\cos26^{\circ}}\\x&\approx20.81\text{ cm}\end{aligned}\)
The rounded cosine value changes the final hundredths. Enter the full cosine calculation before rounding the lengthUsing the longest-side check on two legs
✕\(\displaystyle \begin{aligned}\tan57^{\circ}&=\frac{13.2}{x}\\x&>13.2\end{aligned}\)✓\(\displaystyle \begin{aligned}x&=\frac{13.2}{\tan57^{\circ}}\\x&\approx8.57\text{ cm}\end{aligned}\)
Only the hypotenuse must be longest. Here the unknown is adjacent, and the tangent value is greater than one, so division produces a shorter legDifficulty levels
standardCalculate an unknown hypotenuse or adjacent leg using sine, cosine or tangent
examChoose a side length from three options for the same triangle structures
Questions about this topic
Why do I divide when the unknown is underneath?+
Multiplying both sides by the unknown clears the denominator. The trigonometric value is then a factor of the unknown, so divide both sides by that value
Does division always make the side longer?+
No. A positive divisor below one makes the result larger than the known length. A divisor above one makes it smaller. Tangent values can lie on either side of one
When should I round?+
Round only the final requested length. Keep the full calculator value for the trigonometric ratio and any intermediate lengths
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 problemRead and state three-figure bearingsSolve a navigation problem with bearings