# Apply Exact Trig Values

In this worksheet, students will learn how to calculate the exact values of trigonometry.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Triangle Mensuration

Difficulty level:

### QUESTION 1 of 10

Are you a budding crime scene officer?  Do you love watching CSI and programmes like that?

Did you know that at times trigonometry (trig) is used in crime scenes?

Maybe the investigators want to know the trajectory of something, or the distance, or the angle.

Time for a little story

Once upon a time students studying A level needed to know the exact values of trig.

They may have wanted careers that involved maths at a higher level, or they may just have liked solving puzzles.

Along came someone who said, A level students shouldn't have all the fun, let G.C.S.E students join in.

Is it a crime?  Up to you to decide.

Let's investigate

The exact values of Sin, Cos and Tan that you need to know are those for 0°, 30°, 45° 60° and 90° for each ratio.

There are a few ways of calculating them.

You may have been taught this way.

Taking an equilateral triangle, giving each side a unit of 2, and splitting it into a right-angled triangle as shown.

You would then label the triangle's hypotenuse, opposite and adjacent in order to find the trig values of Sin, Cos or Tan.

e.g Sin 30° = 1/2  Sin 60° = √3/2

or an isosceles triangle to do the same.

If you are secure in labelling the triangles and using the trig ratio then this is absolutely fine.

It can get quite messy having to keep labelling the triangles (opposites and adjacent are not always in the same place)

A different way without having to split and label triangles.

You know the trig values required  0°, 30°, 45° 60° and 90° for each ratio.

You need the numbers 0,1,2,3,4 and be able to simplify square roots if you can.

Exact values of sin

To find exact values of sin √angle ÷ 2

Look at the numbers at the top of the table ranging from 0 to 4 this is assigned to your √

Looking at the example above all the numbers assigned to the angles have been √ ÷ 2 and where possible simplified.

√ can only be simplified to a whole number, for example, √4 = 2

Exact Values of Cos

To find exact values of Cos - √angle ÷ 2

The numbers at the top of the table go from 4 to 0.

What do you notice about the answers?

Exact values of tan

This is slightly different.  Values of sin ÷ values of cos and simplify if possible.

Note that tan 90º can't be done.  If you put 1 ÷ 0 into your calculator it comes up as an error.

During the activities, try to recreate the table for yourself, rather than refer to them as it will help with understanding the calculations.

Before you go any further, find the exact value of Sin 45°?

1/2

√3

√2 /2

1

Find the exact value of Cos 30°.

1/2

√3

√2 /2

1

What is this the exact trig value for?

√3 is the exact value for Sin 30° Cos 60° Tan 30° Tan 60° Cos&30°

True of false?

 True False √2/2 Sin45° √2/2 Cos45°

This is the exact value of Cos 60° and which other trig value?

Cos 30°

Tan 30°

Sin 30°

Cos 45°

How many trig values have the exact value of zero?

What is this the exact trig value for Cos 90°?

√4/2

2/2

0

1

This is the exact trig value for Cos 30° and which other value?

Write your answer in figures without the units.

Have a go at the question below.

## Column B

Sin 0°
0
Cos 45°
impossible
Tan 45°
√2/2
Tan 90°
1

Last question coming up!

## Column B

Cos 0°
1/√3
Cos 30°
√3/2
Tan 30°
1
Sin 60°
√3/2
• Question 1

Before you go any further, find the exact value of Sin 45°?

√2 /2
EDDIE SAYS
The easiest way around these is to draw a table with the headings in first, then you are up and running. Sin 45° has the number 2 assigned to it. √2 ÷ 2 You can't simplify the √ to a whole number
• Question 2

Find the exact value of Cos 30°.

EDDIE SAYS
Remember with Cos the numbers go down from 4 to 0. √4/2 = 2/2 = 1 As long as you know your square roots you should be fine. Who would have thought those lessons learning about square roots would be of benefit to us now.
• Question 3

What is this the exact trig value for?

√3 is the exact value for Sin 30° Cos 60° Tan 30° Tan 60° Cos&30°
EDDIE SAYS
If you divide Sin 60 ° √3/2 and Cos 60° 1/2 you are left with √3 which is Tan 60°.
• Question 4

True of false?

 True False √2/2 Sin45° √2/2 Cos45°
EDDIE SAYS
This is the same for both Sin and Cos 45°, if you look at both Sin and Cos graphs you would see that they both have the same value.
• Question 5

This is the exact value of Cos 60° and which other trig value?

Sin 30°
EDDIE SAYS
How is the table of values coming along? You don't necessarily have to learn the exact values but will need to know how to work them out.
• Question 6

How many trig values have the exact value of zero?

3
EDDIE SAYS
Sin 0°, Cos 90° and Tan 0° Just remember that zero divided by anything is always zero. Don't confuse this with Tan 90° which is 1 ÷ 0 which is impossible.
• Question 7

What is this the exact trig value for Cos 90°?

√4/2
2/2
1
EDDIE SAYS
Any of the above except 0 is correct. Always simplify when you can. Us mathematicians like things simple.
• Question 8

This is the exact trig value for Cos 30° and which other value?

Write your answer in figures without the units.

Sin 60
EDDIE SAYS
For every exact value of Cos there is also one for Sin. This should be repeated in your table, also you should be able to see them on the Sin and Cos graphs.
• Question 9

Have a go at the question below.

## Column B

Sin 0°
0
Cos 45°
√2/2
Tan 45°
1
Tan 90°
impossible
EDDIE SAYS
I hope you are getting used to this now. Why do gardeners dislike maths? It gives them square roots. Roots creep into everything, don't they...
• Question 10

Last question coming up!

## Column B

Cos 0°
1
Cos 30°
√3/2
Tan 30°
1/√3
Sin 60°
√3/2
EDDIE SAYS
I hope you are now able to complete the table of values with ease. Next time you teacher asks you to join the athletic competition you can now say sine me up.....
---- OR ----

Sign up for a £1 trial so you can track and measure your child's progress on this activity.

### What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started

Start your £1 trial today.
Subscribe from £10/month.

• Tuition Partner