# Trigonometry and Bearings

In this worksheet, students use trigonometry to solve bearings questions.

Key stage:  KS 4

Curriculum topic:  Geometry and Measures

Curriculum subtopic:  Interpret and Use Bearings

Difficulty level:

### QUESTION 1 of 10

We can use trigonometry to answer some bearings questions, provided that we have sufficient information.

It is always useful to draw a quick sketch of the situation from the information given in the question.  The diagram does not have to be at all accurate.

Example

A boat is due South of a lighthouse and sails on a bearing of 292º for 51 km until it is due West of the lighthouse.  How far away is it now from the lighthouse (to 3 sig. figs.)?

Draw a sketch using the information given and work out the relevant angles in the right-angled triangle.

Here the interior angle = 360 - 292 = 68º

We want to find the side length LW.

Set up the trig equation to get:

sin68º = LW/51

LW= 51 sin68º

LW = 51 x  0.92718....

LW= 47.3 (3 s.f.)

A boat is due South of a lighthouse and sails on a bearing of 060º for 51 km until it is due East of the lighthouse.

How far away is it now from the lighthouse in km?

(Give your answer to 3 sig. figs. and just write the number)

A boat is due South of a lighthouse and sails on a bearing of 060º for 51 km until it is due East of the lighthouse.

How far South of the lighthouse was it originally in km?

(Give your answer to 3 sig. figs. and just write the number)

A plane is due North of a village.  It flies on a bearing of 104º for 400 km until it is due East of the village.

How far is it now from the village in km?

(Give your answer to 3 sig. figs. and just write the number)

A helicopter is due South of a mountain peak and it flies on a bearing of 340º for 120 km until it is due West of the peak.

How far is it now from the peak in km?

(Give your answer to 3 sig. figs. and just write the number)

Birmingham is due South of Bradford and Hull is due East of Bradford.

Hull is 180 km from Birmingham and its bearing from Birmingham is 032º.

What is the distance from Bradford to Hull in km?

(Give your answer to 3 sig. figs. and just write the number)

Birmingham is due South of Bradford and Hull is due East of Bradford.

Hull is 180 km from Birmingham and its bearing from Birmingham is 032º.

What is the distance from Bradford to Birmingham in km?

(Give your answer to 3 sig. figs. and just write the number)

Bangor is due North of Swansea and Newport is due East of Swansea.

Newport is 200 km from Bangor and its bearing from Bangor is 159º.

What is the distance from Bangor to Swansea in km?

(Give your answer to 3 sig. figs. and just write the number)

Bangor is due North of Swansea and Newport is due East of Swansea.

Newport is 200 km from Bangor and its bearing from Bangor is 159º.

What is the distance in km from Newport to Swansea?

(Give your answer to 3 sig. figs. and just write the number)

Bristol is 228 km due South of Liverpool and Sheffield is due East of Liverpool.

Bristol is on a bearing of 204º from Sheffield.

What is the distance in km from Liverpool to Sheffield?

(Give your answer to 3 sig. figs. and just write the number)

Bristol is 228 km due South of Liverpool and Sheffield is due East of Liverpool.

Bristol is on a bearing of 204º from Sheffield.

What is the distance in km from Bristol to Sheffield?

(Give your answer to 3 sig. figs. and just write the number)

• Question 1

A boat is due South of a lighthouse and sails on a bearing of 060º for 51 km until it is due East of the lighthouse.

How far away is it now from the lighthouse in km?

(Give your answer to 3 sig. figs. and just write the number)

44.2
EDDIE SAYS
51sin60°
• Question 2

A boat is due South of a lighthouse and sails on a bearing of 060º for 51 km until it is due East of the lighthouse.

How far South of the lighthouse was it originally in km?

(Give your answer to 3 sig. figs. and just write the number)

25.5
EDDIE SAYS
51cos60°
• Question 3

A plane is due North of a village.  It flies on a bearing of 104º for 400 km until it is due East of the village.

How far is it now from the village in km?

(Give your answer to 3 sig. figs. and just write the number)

388
EDDIE SAYS
400sin76°
• Question 4

A helicopter is due South of a mountain peak and it flies on a bearing of 340º for 120 km until it is due West of the peak.

How far is it now from the peak in km?

(Give your answer to 3 sig. figs. and just write the number)

41.0
EDDIE SAYS
120sin20°
• Question 5

Birmingham is due South of Bradford and Hull is due East of Bradford.

Hull is 180 km from Birmingham and its bearing from Birmingham is 032º.

What is the distance from Bradford to Hull in km?

(Give your answer to 3 sig. figs. and just write the number)

95.4
EDDIE SAYS
180sin32°
• Question 6

Birmingham is due South of Bradford and Hull is due East of Bradford.

Hull is 180 km from Birmingham and its bearing from Birmingham is 032º.

What is the distance from Bradford to Birmingham in km?

(Give your answer to 3 sig. figs. and just write the number)

153
EDDIE SAYS
180cos32°
• Question 7

Bangor is due North of Swansea and Newport is due East of Swansea.

Newport is 200 km from Bangor and its bearing from Bangor is 159º.

What is the distance from Bangor to Swansea in km?

(Give your answer to 3 sig. figs. and just write the number)

187
EDDIE SAYS
200cos21°
• Question 8

Bangor is due North of Swansea and Newport is due East of Swansea.

Newport is 200 km from Bangor and its bearing from Bangor is 159º.

What is the distance in km from Newport to Swansea?

(Give your answer to 3 sig. figs. and just write the number)

71.7
EDDIE SAYS
200sin21°
• Question 9

Bristol is 228 km due South of Liverpool and Sheffield is due East of Liverpool.

Bristol is on a bearing of 204º from Sheffield.

What is the distance in km from Liverpool to Sheffield?

(Give your answer to 3 sig. figs. and just write the number)

102
EDDIE SAYS
228tan24°
• Question 10

Bristol is 228 km due South of Liverpool and Sheffield is due East of Liverpool.

Bristol is on a bearing of 204º from Sheffield.

What is the distance in km from Bristol to Sheffield?

(Give your answer to 3 sig. figs. and just write the number)

250
EDDIE SAYS
228/cos24°
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