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Find a Side Using the Cosine Rule

In this worksheet, students will apply the cosine rule to find out lengths of missing sides of a non-right-angled triangle.

'Find a Side Using the Cosine Rule' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Triangle Mensuration

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Ever wondered what these guys are going at the side of the road?

They are calculating distances  between one point and another using trigonometry.

 

Surveyor

Trigonometry is the study of angles and dimensions of triangles.

Unfortunately as schools tend not to have this specialist equipment, we are going to have to make do with the old fashioned method of pen and paper.

The cosine rule is used to calculate lengths in a non right angled triangle.  And here it is.

No... how on earth do you choose which one?

You may have been taught to label the triangle to find a, b or c and get really confused by it.

Try this, and you won't have to worry about labelling the triangle.

 

cosine rule simplified

 

You will need a scientific calculator in order to get the cos button. Make sure it is set to degrees.

Example 1

 

cosine rule example

Explanation

We are given two side lengths and the trapped angle.

 

Square one side, and add the square of the other side.

Subtract 2 x side 1 x side 2 x Cos63(angle)

Square root your answer.

 

 

Example 2

cosine rule example

       Can you see that all you need are the two sides with the trapped angle? 

                                  When you use the angle, you will always use the COS button.

                                  As long as you remember the general formula you need not worry about a,b, or c                      

            

cosine rule

Answer has been rounded to 3 significant figures.

20.5 cm

16.1 cm

6.24cm

39.2 cm

cosine find a side

 7.05 cm6.50 cm6.05cm
The missing side length =

cosine rule find a side

Round your answer to 3 significant figures (3 s.f)

 7.05 cm6.50 cm6.05cm
The missing side length =

cosine rule find a side

Calculate the missing length.

Round your answer to 3 significant figures (s.f)

Write your answer in figures without the units.

cosine rule find a side

What is the length of the missing side?

The answer has been rounded to 3 significant figures (3 s.f)

10.0 cm

10.2cm

10.3 cm

10.1 cm

 

 

 8.95cm8.92cm8.91 cm
The missing side length is

cosine rule find a side

Round your answer to 3 significant figures (3 s.f)

 

 8.95cm8.92cm8.91 cm
The missing side length is

cosine rule find a side

What is the length of the missing side.

Round your answer to 3 significant figures.

Write your answer in figures without the units.

Round your answer to 3 significant figures (3 s.f)

cosine find a rule

 

Round your answer to 3 significant figures( 3 s.f)

14.2 cm

14.5 cm

14.1 cm

14.0 cm

  • Question 1

cosine rule

Answer has been rounded to 3 significant figures.

CORRECT ANSWER
6.24cm
EDDIE SAYS
Was it that scary? How did you get on? Did you remember that you need the sides and the trapped angle. 5² + 7² - 2 x 5 x 7 x Cos60 = 39 √39 = 6.24 cm( to 3 s.f) Did you remember the √ ? this is the bit students often forget.
  • Question 2

cosine find a side

CORRECT ANSWER
 7.05 cm6.50 cm6.05cm
The missing side length =
EDDIE SAYS
Can you see the resemblance to Pythagoras theorem? Finding a side length means squaring sides. Once you are comfortable with this it may be that you just need to learn the second part of the formula 8² + 7² - 2 x 7 x 8 x Cos47 = 36.61 √36.61 = 6.05 cm( to 3 s.f)
  • Question 3

cosine rule find a side

Round your answer to 3 significant figures (3 s.f)

CORRECT ANSWER
EDDIE SAYS
As there are so many mathematical formulas to remember, this formula is often given to you in text books or at the front of exam papers. As long as you recognise is at using the sides and trapped angle it should take the pressure off you. With practice it soon becomes second nature. 5² + 8² - 2 x 5 x 8 x Cos42 = 29.54 √29.54 = 5.44 cm( to 3 s.f)
  • Question 4

cosine rule find a side

Calculate the missing length.

Round your answer to 3 significant figures (s.f)

Write your answer in figures without the units.

CORRECT ANSWER
9.25
EDDIE SAYS
Is this becoming a little easier now? 10² + 11² - 2 x 10 x 11 x Cos52 = 85.554 √85.554= 9.25 cm( to 3 s.f)
  • Question 5

cosine rule find a side

What is the length of the missing side?

The answer has been rounded to 3 significant figures (3 s.f)

CORRECT ANSWER
10.1 cm
EDDIE SAYS
It is always good to write out the formula first and then substitute as it leaves little room for error. Don't try to do this in your head, there is to much information to remember and you could get muddled up. 8² + 5² - 2 x 8 x 5 x Cos100 = 102.98 √102.89= 10.1 cm( to 3 s.f)
  • Question 6

 

 

CORRECT ANSWER
 8.95cm8.92cm8.91 cm
The missing side length is
EDDIE SAYS
Think of using the cosine rule like a triangle. Three things to do 1. Substitute in the numbers 2.Calculate 3. Square root 7² + 4² - 2 x 7 x 4 x Cos105 = 79.493 √79.493 = 8.92cm( to 3 s.f)
  • Question 7

cosine rule find a side

Round your answer to 3 significant figures (3 s.f)

 

CORRECT ANSWER
EDDIE SAYS
Hows it going. Practice, practice, practice is all it takes. That is the beauty of a formula, it gives us step by step instructions. 3² + 3² - 2 x 3 x 3 x Cos42 = 4.623 √4.623= 2.15 cm( to 3 s.f) A good way to check if you are correct is to check that the length is sensible when compared to the other two sides.
  • Question 8

cosine rule find a side

What is the length of the missing side.

Round your answer to 3 significant figures.

Write your answer in figures without the units.

CORRECT ANSWER
10.3
EDDIE SAYS
8² + 11² - 2 x 8 x 11 x Cos63= 105.09 √105.09= 10.3 cm( to 3 s.f) It is important to be able to round to significant figures as these type of questions often demand it. Yes I know ...something else to practice
  • Question 9

Round your answer to 3 significant figures (3 s.f)

CORRECT ANSWER
EDDIE SAYS
Come on admit it, you are having fun now. 7² + 4² - 2 x 7 x 4 x Cos57= 34.500 √34.500 = 5.87 cm( to 3 s.f)
  • Question 10

cosine find a rule

 

Round your answer to 3 significant figures( 3 s.f)

CORRECT ANSWER
14.1 cm
EDDIE SAYS
You are now ready to go and help those surveyors. One last check before you go. 9² + 12² - 2 x 9 x 12 x Cos83 = 198.67 √198.67= 14.1 cm( to 3 s.f)
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