Finding an Unknown Side
Learn to find an unknown side of a right-angled triangle for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-04). Once the sides are labelled from the given angle, a single trigonometric ratio links the side you know to the side you want.
You will use the trigonometric ratios to find missing side lengths, with the angle given in degrees or in degrees and minutes: choose sine, cosine or tangent, multiply or divide to solve, and keep your calculator in degree mode — for practical problems such as the height of a tree, a ladder or the length of a ramp.
Theory
To find an unknown side of a right-angled triangle, label the sides from the given angle, choose sine, cosine or tangent, write the equation and solve it on your calculator. This Year 12 Mathematics Standard 1 (NSW) guide covers \(x\) on the top (multiply), \(x\) on the bottom (divide) and angles in degrees and minutes.
To find an unknown side, first label the sides from the given angle: hypotenuse, opposite and adjacent. Then choose the ratio that uses the side you know and the side you want: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\) or \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\).
If \(x\) is on the top of the fraction, multiply: \(\sin32^\circ=\dfrac{x}{14}\Rightarrow x=14\sin32^\circ\). If \(x\) is on the bottom, divide: \(\sin32^\circ=\dfrac{6}{x}\Rightarrow x=\dfrac{6}{\sin32^\circ}\).
Angles may be given in degrees and minutes: \(60'=1^\circ\), so \(42^\circ30'=42.5^\circ\). Enter the angle with the calculator's degrees–minutes–seconds key (or as a decimal), with the calculator in degree mode, and round only the final answer. Uses include ladder heights, ramp lengths and the height of a tree from a measured angle.
The three ratios, with \(\theta\) the given angle:
\(x\) on the top: multiply.
\(x\) on the bottom: divide.
Degrees and minutes:
How to find an unknown side
- Label the sides from the given angle: hyp, opp, adj.
- Choose the ratio (SOH CAH TOA) with the known side and the unknown side, and write the equation.
- Solve: multiply if \(x\) is on top, divide if \(x\) is on the bottom. Use degree mode, then round only the final answer and include units.
Sketch the triangle. The hypotenuse (\(16\) cm) is known and \(x\) is opposite \(27^\circ\), so use sine.
| \(\sin27^\circ\) | \(=\) | \(\dfrac{x}{16}\) |
| \(x\) | \(=\) | \(16\times\sin27^\circ\) |
| \(x\) | \(=\) | \(7.2638\ldots\) |
\(x\approx\) \(7.26\) cm.
Sketch the triangle. The adjacent side (\(12\) m) is known and \(x\) is opposite, so use tangent. Enter \(46^\circ15'\) as \(46.25^\circ\).
| \(\tan46^\circ15'\) | \(=\) | \(\dfrac{x}{12}\) |
| \(x\) | \(=\) | \(12\times\tan46^\circ15'\) |
| \(x\) | \(=\) | \(12.5353\ldots\) |
\(x\approx\) \(12.54\) m.
Sketch the triangle. The adjacent side (\(6.5\) m) is known and \(x\) is the hypotenuse, so use cosine and rearrange.
| \(\cos34^\circ\) | \(=\) | \(\dfrac{6.5}{x}\) |
| \(x\times\cos34^\circ\) | \(=\) | \(6.5\) |
| \(x\) | \(=\) | \(\dfrac{6.5}{\cos34^\circ}\) |
| \(x\) | \(=\) | \(7.8404\ldots\) |
\(x\approx\) \(7.84\) m (longer than \(6.5\) m, as it should be).
Sketch the ladder. It is the hypotenuse; \(h\) is opposite (sine) and \(d\) is adjacent (cosine).
| \(\text{(i) }h\) | \(=\) | \(5.5\times\sin64^\circ40'\) |
| \(\) | \(=\) | \(4.9710\ldots\) |
| \(\text{(ii) }d\) | \(=\) | \(5.5\times\cos64^\circ40'\) |
| \(\) | \(=\) | \(2.3533\ldots\) |
(i) \(h\approx\) \(4.97\) m, (ii) \(d\approx\) \(2.35\) m.
Common pitfalls
Frequently asked questions
How do you find a missing side of a right-angled triangle using trigonometry?
Label the sides from the given angle, choose sine, cosine or tangent so the ratio uses the known side and the unknown side, write the equation, then solve it on your calculator in degree mode.
When do you multiply and when do you divide in trigonometry?
If the unknown side x is on the top of the fraction, multiply: sin 32 degrees = x over 14 gives x = 14 sin 32 degrees. If x is on the bottom, divide: sin 32 degrees = 6 over x gives x = 6 divided by sin 32 degrees.
How do you enter degrees and minutes on a calculator?
Use the degrees-minutes-seconds key, or convert to a decimal first. There are 60 minutes in a degree, so 46 degrees 15 minutes is 46.25 degrees.
Why is my calculator giving the wrong answer for sin?
It is probably not in degree mode. Check that sin 30 degrees gives 0.5; if it does not, change the calculator to degrees.
How do you find the height a ladder reaches up a wall?
The ladder is the hypotenuse and the height is opposite the angle with the ground, so height = ladder length times the sine of the angle. A 5.5 m ladder at 64 degrees 40 minutes reaches about 4.97 m.