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Find an Angle Using the Cosine Rule

In this worksheet, students will apply the cosine rule to find out the size of angles within a non-right-angled triangle.

'Find an Angle 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 it's like to dangle outside of the space station to go to work.

 

astronaut

 

The robotic arms are operated by controlling the angles of its joints.  Calculating the final position of the astronaut involves trigonometry, one aspect of which is finding angles.

 

Things have come a long way since man first landed on the moon, who would have thought that studying angles could get us this job.

Did your careers advisor mention this as a future job?  I suspect not.

In preparation for our chosen career, we had better get to work.

 

The cosine rule is used to find angles in a non-right angled triangle.  And here it is.

 

Three for the price of one, not always the best deal!

 

Try this, and you won't have to worry about what a, b or c actually is.

 

cosine rule simplified

 

This looks similar to that of finding a side using the cosine rule.

Make sure your calculator is set to degrees.

 

Example 1

cosine rule find angle example

Explanation

We are given two side lengths and the trapped angle.

 

Square one side, (6²) and add the square(5²) of the other side.

Subtract the square of the side that is opposite the angle. (7²)

Then multiply 2 by the side lengths next to the angle. e.g 2 x 6 x 5

Divide your answers.

As you want to find an angle you need Cos-1 

 

 

Example 2

cosine rule find an angle example

 

NOTE.  You can see here that you have a negative Cos  - 0.276. This means that the angle you convert to will be obtuse i.e over 90°  This is a good way to check if your answer is sensible.

 

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

When want to convert to find the angle, you will always use the COS-1 button.

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

            

cosine rule find an angle

Calculate the value of x in this diagram.

 

Round your answer to 3 significant figures.

32.3°

28.6°

35.4°

25.2°

Find the value of the angle represented by x in this diagram: 

 71.4°78.5°60.2°
Angle x =

Find the value of the angle represented by x in this diagram: 

 

cosine rule find an angle

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

 71.4°78.5°60.2°
Angle x =

cosine rule find an angle

Calculate the value of x.

 

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

 71.4°78.5°60.2°
Angle x =

cosine rule find an angle

Find the value of x.

 

Round your answer to 3 significant figures.

23.9°

23.7°

23.0°

23.1°

Find the value of the angle represented by x in this diagram: 

cosine rule find an angle

 

 

 225°120°103°
Angle x is

Find the value of the angle represented by x in this diagram: 

 

cosine rule find an angle

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

 

 225°120°103°
Angle x is

Find the value of the angle represented by x in this diagram: 

 

cosine rule find angle

Round your answer to 3 significant figures.

 225°120°103°
Angle x is

Find the value of the angle represented by x in this diagram: 

cosine rule find an angle

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

 225°120°103°
Angle x is

Find the value of the angle represented by x in this diagram:

cosine rule find an angle

 

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

34.9°

82.3°

41.7°

50.6°

  • Question 1

cosine rule find an angle

Calculate the value of x in this diagram.

 

Round your answer to 3 significant figures.

CORRECT ANSWER
25.2°
EDDIE SAYS
Was your first mission a success? Did you remember that you need the sides and the trapped angle? 7² + 9² - 4² = 114 2 x 7 x 9 = 126 114 ÷ 126 = 0.904 Cos-1 0.904 = 25.2° (to 3 s.f)
  • Question 2

Find the value of the angle represented by x in this diagram: 

CORRECT ANSWER
 71.4°78.5°60.2°
Angle x =
EDDIE SAYS
What time do astronauts eat? At launch time! Are we ready on the launchpad for this one? 6² + 6² - 7² = 23 2 x 6 x 6 =72 23 ÷ 72 = 0.3194.. Cos-1 0.3194.. = 71.4° (to 3 s.f)
  • Question 3

Find the value of the angle represented by x in this diagram: 

 

cosine rule find an angle

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

CORRECT ANSWER
EDDIE SAYS
Did you notice the negative cosine? How could you check your answer was sensible? Yes, a negative cosine means the angle is obtuse. So the answer here makes sense. Let's work it out: 5² + 9² - 12² = -38 2 × 5 × 9 = 90 -38 ÷ 90 = - 0.4222 Cos-1 0.4222 = 114.974° Which is 115° when rounded to 3 significant figures.
  • Question 4

cosine rule find an angle

Calculate the value of x.

 

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

CORRECT ANSWER
EDDIE SAYS
Are these questions driving you mad or are you getting the hang of the formula? Hang in there, practice makes perfect! Let's apply our formula once more: 6² + 8² - 11² = -21 2 × 6 × 8 = 96 -21 ÷ 96 = -0.21875 Cos-1 0.21875 = 102.6356...° If we round this to 3 significant figures, our answer should be 103° - how did you get on?
  • Question 5

cosine rule find an angle

Find the value of x.

 

Round your answer to 3 significant figures.

CORRECT ANSWER
23.1°
EDDIE SAYS
Formulas really are helpful aren't they. Don't try to do this in your head, there is to much information to remember and you could get muddled up. 10² +10² - 4² = 184 2 x 10 x 10 = 200 184 ÷ 200 = 0.92 Cos-1 0.92 = 23.1° (to 3 s.f)
  • Question 6

Find the value of the angle represented by x in this diagram: 

cosine rule find an angle

 

 

CORRECT ANSWER
 225°120°103°
Angle x is
EDDIE SAYS
What do you do when you get the cosine rule? You rocket... are you rocking it yet? Yes or no, either way keep practising! 12² +5² - 14² = -27 2 × 12 × 5 = 120 -27 ÷ 120 = -0.225 Cos-1 0.225 = 103.0028...° If we round this number to 3 s.f., we reach 103°.
  • Question 7

Find the value of the angle represented by x in this diagram: 

 

cosine rule find an angle

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

 

CORRECT ANSWER
EDDIE SAYS
Are you dancing the moonwalk yet? Let's hope you are not getting the steps muddled up! 7² +5² - 4² = 58 2 × 7 × 5 = 70 58 ÷ 70 = 0.828... Cos-1 0.828 = 34.047...° Rounded to 3 s.f. = 34.0° That was a little sneaky, wasn't it? Hopefully you remembered to use three significant figures as the question requested, rather than two!
  • Question 8

Find the value of the angle represented by x in this diagram: 

 

cosine rule find angle

Round your answer to 3 significant figures.

CORRECT ANSWER
EDDIE SAYS
5.5² +7.5² - 6.5² = 44.25 2 × 5.5 × 7.5 = 82.5 44.25 ÷ 82.5 = 0.5363 Cos-1 0.5363 = 57.6° (to 3 s.f) It is important to be able to round to significant figures as these type of questions often demand it. Yes we know... one more thing to practise!
  • Question 9

Find the value of the angle represented by x in this diagram: 

cosine rule find an angle

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

CORRECT ANSWER
EDDIE SAYS
Are you out of this world with excitement yet? Using the cosine rule is hopefully not too tricky after all! 8² +7² - 14² = -83 2 × 8 × 7 = 112 -83 ÷ 112 = -0.741... Cos-1 0.741... = 137.822...° To 3 s.f. = 138°
  • Question 10

Find the value of the angle represented by x in this diagram:

cosine rule find an angle

 

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

CORRECT ANSWER
34.9°
EDDIE SAYS
We apollogise for all the space jokes. But come on, how else can we liven up the cosine rule? 5² +5² - 3² = 41 2 × 5 × 5 = 50 41 ÷ 50 = 0.82 Cos-1 0.82 = 34.9° (to 3 s.f)
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