Trigonometric Ratios
Finding Lengths in Right Angled Triangles

When one angle and the length of one side are known, it is possible to find the lengths of other sides in the same triangle, by using the sine, cosine or tangent formula.

For example,

sin50° =   ⇒   x = 12 sin50°

cos50° =   ⇒   y = 12 cos50°

Worked Examples

1

Find the length of the side marked x in the triangle shown.

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In this triangle,

hypotenuse = 20 cm

opposite = x cm

Choose sine because it involves hypotenuse and opposite.

Using sinθ =

gives sin70° =

To obtain x, multiply both sides of this equation by 20, which gives

20 sin70° = x
or
x = 20 sin70°
= 18.8 cm (to 1 decimal place)

This value is obtained using a calculator.
2

Find the length of the side marked x in the triangle.

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In this triangle,

opposite = x

adjacent = 8 metres

Use tangent because it involves the opposite and adjacent.

Using tanθ =

gives tan40° =

Multiplying both sides by 8 gives

8 tan40° = x
or
x = 8 tan40°
= 6.7 metres (to 1 decimal place)
3

Find the length marked x in the triangle.

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In this triangle, tan42° = =

Multiplying by x gives

x tan42° = 10

and dividing by tan42° gives

x =
=
= 11.1 metres (to 1 decimal place)
4

Find the length of the hypotenuse, marked x, in the triangle.

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In this triangle,

hypotenuse = x

opposite = 10 cm

Use sine because it involves hypotenuse and opposite.

Using sinθ =

gives sin28° =

where x is the length of the hypotenuse.

Multiplying both sides by x gives

x sin28° = 10,

then dividing both sides by sin 28° gives

x =
= 21.3 cm (to 1 decimal place)
5

The diagram above, not drawn to scale, represents one face of the roof of a house in the shape of a parallelogram EFGH. Angle EFI = 40° and EF = 8 m. EI represents a rafter placed perpendicular to FG such that IG = 5 m.

Calculate, giving your answers to 3 significant figures,

(a)

the length of FI

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FI = 8 cos40 ≈ 6.13 m
(b)

the length of EI

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EI = 8 sin40 ≈ 5.14 m
(c)

the area of EFGH.

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Area of EFGH = FG × EI
= (5 + 6.13) × 5.14
= 11.13 × 5.14
≈ 57.2 m2

Exercises

On this page round your answers to 2 decimal places where necessary.

Find the length of the side marked x in each triangle.

(a)
x = cm
(b)
x = cm
(c)
x = cm
(d)
x = cm
(e)
x = cm
(f)
x = cm
(g)
x = cm
(h)
x = cm
(i)
x = m
(j)
x = m
(k)
x = m
(l)
x = m
(m)
x = m
(n)
x = m
(o)
x = m

A guy rope is attached to a tent peg and the top of a tent pole so that the angle between the peg and the bottom of the pole is 60°.

(a)

Find the height of the pole if the peg is 1 metre from the bottom of the pole.

m
(b)

If the length of the rope is 1.4 metres, find the height of the pole.

m
(c)

Find the distance of the peg from the base of the pole if the length of the guy rope is 2 metres.

m

A child is on a swing in a park. The highest position that she reaches is as shown.

Find the height of the swing seat above the ground in this position.

m

A ship sails 400 km on a bearing of 075° .

(a)

How far east has the ship sailed?

km (to 1 d.p.)
(b)

How far north has the ship sailed?

km (to 1 d.p.)

An aeroplane flies 120 km on a bearing of 210° .

(a)

How far south has the aeroplane flown?

km (to 1 d.p.)
(b)

How far west has the aeroplane flown?

km

Find the length of the side marked x in each triangle.

(a)
x = cm
(b)
x = cm
(c)
x = cm
(d)
x = m
(e)
x = m
(f)
x = m
(g)
x = cm
(h)
x = cm
(i)
x = m

A snooker ball rests against the side cushion of a snooker table. It is hit so that it moves at 40° to the side of the table. How far does the ball travel before it hits the cushion on the other side of the table?

cm (to 1 d.p.)

In the figure shown, calculate

(a)
the length of BD. cm
(b)
the length of BC. cm