Finding Unknown Side
Once an angle and one side of a right-angled triangle are known, a trigonometric ratio finds another side. In this NSW Year 9 Core maths subtopic the unknown is always on the top of the ratio, so the last step is a multiplication, as in \(x = 12\cos 35^\circ\). It comes under NESA outcome MA5‑TRG‑C‑01.
You will label the sides from the given angle, choose sine, cosine or tangent from the two sides involved, substitute and multiply, and round to one or two decimal places. You will also work with angles given in degrees and minutes, triangles described only by their vertices, and worded problems such as the height of a boat's mast, the rise of a wheelchair ramp and how far up a wall a ladder reaches. It is part of Year 9 Core Trigonometry.
Theory
To find an unknown side of a right-angled triangle with trigonometry, label the sides from the given angle, choose the ratio that uses the known side and the wanted side, and multiply. This Year 9 maths page covers sin, cos and tan, angles in degrees and minutes, triangles named by their vertices, and worded problems.
When one angle (besides the right angle) and one side of a right-angled triangle are known, the other sides can be found with a trigonometric ratio.
The sides are labelled from the given angle: the hypotenuse faces the right angle, the opposite faces the angle, and the adjacent is next to it.
Choose the ratio that contains BOTH the side you know and the side you want. On this page the unknown side is on top of the fraction, so the last step is a multiplication.
Angles can be given in degrees and minutes. One degree is \(60\) minutes, so \(48^\circ 24' = 48.4^\circ\).
The three ratios:
With the unknown on top, multiply both sides by the bottom:
How to find an unknown side
- Sketch the triangle and mark the right angle, the given angle, the known side and the wanted side.
- Label hyp, opp and adj from the given angle.
- Choose the ratio that uses the known side and the wanted side.
- Substitute and multiply to get the unknown on its own. Change any minutes into degrees first.
- Round as asked, and check a shorter side is shorter than the hypotenuse.
From the \(35^\circ\) angle the sides are the opposite \(x\) and the hypotenuse \(18\), so use sin. The unknown is on top, so multiply:
| \(\sin 35^\circ\) | \(=\) | \(\dfrac{x}{18}\) |
| \(x\) | \(=\) | \(18 \sin 35^\circ\) |
| \(\) | \(=\) | \(10.324\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{10.3} \text{ cm}\) |
\(10.3 < 18\), as a shorter side must be.
The right angle is at \(L\), so \(KM\) is the hypotenuse. From the angle at \(K\), \(KL\) runs to the right angle: it is the adjacent side. Use cos:
| \(\cos 62^\circ\) | \(=\) | \(\dfrac{KL}{26}\) |
| \(KL\) | \(=\) | \(26 \cos 62^\circ\) |
| \(\) | \(=\) | \(12.206\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{12.2} \text{ m}\) |
Sketch the triangle from the words first; it makes the labelling obvious.
Opposite and adjacent: no hypotenuse, so use tan. Change the minutes first: \(24' = \dfrac{24}{60} = 0.4^\circ\).
| \(\tan 48.4^\circ\) | \(=\) | \(\dfrac{p}{11}\) |
| \(p\) | \(=\) | \(11 \tan 48.4^\circ\) |
| \(\) | \(=\) | \(12.389\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{12.4} \text{ cm}\) |
Or enter \(48^\circ 24'\) directly with the degrees-minutes key.
The angle is at the top, between the ladder and the wall. From there, the ladder is the hypotenuse and the distance \(d\) along the ground is opposite. Use sin:
| \(\sin 18^\circ\) | \(=\) | \(\dfrac{d}{6}\) |
| \(d\) | \(=\) | \(6 \sin 18^\circ\) |
| \(\) | \(=\) | \(1.854\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{1.85} \text{ m}\) |
If the angle had been measured from the ground, \(d\) would be the adjacent side instead.
Common pitfalls
Frequently asked questions
How do you find a missing side with trigonometry?
Label the sides from the given angle, choose the ratio that uses the known side and the wanted side, substitute, and solve. If the unknown is on top, multiply.
How do I know whether to use sin, cos or tan?
Look at the two sides in the question. Opposite and hypotenuse means sin, adjacent and hypotenuse means cos, opposite and adjacent means tan.
How do you use degrees and minutes in trigonometry?
Divide the minutes by \(60\) and add them to the degrees, or use the degrees-minutes key on the calculator. \(48^\circ 24' = 48.4^\circ\).
When do you multiply and when do you divide?
If the unknown side is on top of the ratio, multiply. If it is on the bottom, divide. That second case is covered in Finding a Side (Unknown Denominator).