 # 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. 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:   ### QUESTION 1 of 10

Ever wondered what its like to dangle outside of the space station to go to work. Cool or scary, I don't know.

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 best deal!

Try this, and you won't have to worry about what a, b or c actually is. This looks similar to that of finding a side using the cosine rule.

Make sure you calculator is set to degrees.

Example 1 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

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

Example 2 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 Calculate the value of x.

Answer has been rounded to 3 significant figures.

32.3°

28.6°

35.4°

25.2° 71.4° 78.5° 60.2° Angle x = 71.4° 78.5° 60.2° Angle x = Calculate the value of x. Find the value of x.

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

23.9°

23.7°

23.0°

23.1° 225° 120° ; 103° Angle x is 225° 120° ; 103° Angle x is   34.9°

82.3°

41.7°

50.6°

• Question 1 Calculate the value of x.

Answer has been rounded to 3 significant figures.

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 71.4° 78.5° 60.2° Angle x =
EDDIE SAYS
What time do astronauts eat? At launch time. Area we ready on the launch pad 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 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. 5² + 9² - 12² = -38 2 x 5 x 9 = 90 -38 ÷ 90 = 0.4222 Cos-1 0.4222 = 114.8° (to 3 s.f)
• Question 4 Calculate the value of x.

102.6
EDDIE SAYS
What do you call a tick on the moon? A luna tick. Is this driving you mad or are you getting the hang of the formula. Hang in there, practice makes perfect. 6² + 8² - 11² = -21 2 x 6 x 8 = 96 21 ÷ 96 = 0.218 Cos-1 0.218 = 102.6 °(to 3 s.f)
• Question 5 Find the value of x.

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

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 6 225° 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 practicing. 12² +5² - 14² = -27 2 x 12 x 5 = 120 -27 ÷ 120 = 0.225 Cos-1 0.225 = 103° (to 3 s.f)
• Question 7 EDDIE SAYS
Are you dancing the moonwalk yet? Lets hope you are not getting the steps muddled up. 7² +5² - 4² = 58 2 x 7 x 5 = 70 58 ÷ 70 = 0.828 Cos-1 0.828 = 34.0° (to 3 s.f)
• Question 8 57.6
EDDIE SAYS
5.5² +7.5² - 6.5² = 44.25 2 x 5.5 x 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 I know ...something else to practice
• Question 9 EDDIE SAYS
Are you out of this world with excitement yet. Using the cosine rule is not that bad after all. 8² +7² - 14² = -83 2 x 8 x 7 = 112 -83 ÷ 112 = -0.7410 Cos-1 0.7410 = 137° (to 3 s.f)
• Question 10 34.9°
EDDIE SAYS
I apollo.gize for the space jokes. But come on, how else am I going to liven up the cosine rule? 5² +5² - 3² = 41 2 x 5 x 5 = 50 41 ÷ 50 = 0.82 Cos-1 0.82 = 34.9° (to 3 s.f)
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