Using Trigonometry to Find a Missing Side
Prior Knowledge:
• Using Pythagoras’ theorem.
• Substitution into a formula.
• Rounding to decimal places and significant figures.
• Rearranging a formula.
Trigonometry can be used to find a missing side in a right-angled triangle, where we also know an
angle. It uses the three trigonometric functions: sine(x), cosine(x) and tangent(x), where x would
be replaced by the angle you are interested in.
You might have seen them on your calculator as sin, cos and tan. Each of these functions has a
formula:
opposite
sin(x) = hypotenuse
adjacent
cos(x) = hypotenuse
opposite
tan(x) = adjacent
Opposite, adjacent and hypotenuse are the names of the three sides of a right-angled triangle.
You should recognise the hypotenuse from your work on Pythagoras’ theorem. It is opposite the
right angle and is always the longest side.
Opposite and adjacent sides are not fixed; they depend on the angle you’re interested in. The
opposite side is farthest from the angle (but not the hypotenuse). The adjacent side is next to the
angle (but, again, not the hypotenuse).
Hy
po
te
Hy
po
Opposite
Adjacent
50°
nu
se
te
nu
Adjacent
Opposite
se
40°
The two triangles above are identical but, because we are interested in different angles each time,
the adjacent and opposite sides change.
You also need to be able to remember the three formulae above; they won’t be given to you in
an exam. A common way of doing this is to remember them as a word: SOHCAHTOA. This can be
written out to help rearrange the formulae:
O
S×H
A
C×H
O
T×A
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Using Trigonometry to Find a Missing Side
Example 1
Find the length of the missing side, k.
k
10cm
30°
Our first step is to label the sides:
H
O
10cm
k
30°
A
For this question, we are interested in the opposite (because we know its length) and the
hypotenuse (because we want to know its length). In this case, we’re not interested in the adjacent.
Therefore, we need to use the trigonometric function that contains the opposite and the
hypotenuse:
opposite
O
sin(x) = hypotenuse = H
We want to find the hypotenuse, so we need to rearrange our equation:
O
H = sin(x)
Hint:
O
You can also do this step by using the formula triangle: S × H
Cover H – this tells you that you need to calculate
O
.
S
Substitute the values:
10
H = sin(30°)
(Use the sin button on your calculator.)
H = 20cm
(Don’t forget units.)
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Using Trigonometry to Find a Missing Side
Example 2
40°
A 5m
y
O
Here, the sides are already labelled. We’re interested in the adjacent and the opposite so we use:
opposite
tan(x) = adjacent
which gives us:
O = A × tan(x)
(Rearrange your formula.)
y = 5 × tan(40°)
(Substitute your values.)
y = 4.1954…
(Use your calculator.)
y = 4.2m
(Round to 1d.p.)
Your Turn
1. Label each side adjacent, opposite or hypotenuse, relative to the angle given:
a.
b.
25°
30°
c.
45°
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Using Trigonometry to Find a Missing Side
2. In each question, find the missing length, marked k. Give your answers correct to 1d.p.
a.
b.
k
8cm
10cm
40°
c.
k
38°
d.
32°
12cm
k
k
38°
7m
e.
k
67°
5m
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Using Trigonometry to Find a Missing Side
3.
a. A rectangle is made out of steel beams. To make it stronger, a diagonal beam is placed across
the middle. The diagonal beam makes an angle of 35° with the longer side, which is 8m long.
How long is the diagonal beam? Give your answer correct to 1d.p.
(Hint: in a written question, start by drawing a diagram and finding the right-angled triangle.)
b. Bob needs to clean a window on the third floor and his ladder must reach at least 7m above
the ground for him to reach it. If his ladder is 8m long and makes an angle of 60° with the
ground (which is horizontal), is it long enough to reach the window?
4. Find the length of the missing side, w. Give your answer correct to 2s.f.
w
50°
30cm
40°
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Using Trigonometry to Find a Missing Side
5. A mnemonic is a pattern or words that help you remember something.
For example, Sydney opera house can always hold thousands of Australians.
Think of your own mnemonic to remember the trigonometric ratios.
Challenge:
Find the length of the missing side w, correct to 2s.f.
10m
50°
w
6m
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