1. trigonometry Radian measure
Sin cos and tan of angles measured in radians

2. Radian measure A
An angle is a measure of turn and so far we have measured angles in degrees
120º O
< AOB = 120º
The circle was divided into 360 sectors and the angle at the centre of each sector is 1º B
360 could have been chosen because it has many factors 1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360 The circle can then be divided into many sectors where the angle at the centre is a whole number
So 360 was a convenient number to choose.

3. The radian 1 radian We therefore measure angles in radians
For more advanced maths the measure of turn must be a real number and not convenient number
We therefore measure angles in radians
The radian A r
OA = OB = r the radius of the circle centre O r B
When arc AB also = r r O
Then angle AOB is measured to be
1 radian
We can see that this is close to 60º
Like an equilateral triangle

4. Comparing radians and degreesB = r O r =
1 radian =
 radians =
180 º

5. Express as a multiple of
We should now be able to change degrees into radians using
180 º =  radians
90 º =
radians
The fraction the angle 90 is of 180 is the same as the fraction the angle in radians is of  that is /2
Similarly 45º =  /4
60º = /3
30º = /6
Calculating angles > 90
Express as a multiple of
30, 45 or 60
135º = 3 X 45º
= 3 X /4
=3/4
In most of our work with radians the angle is expressed as a multiple of  . So we should be able to convert easily between degrees and radians or memorise them

6. degrees 30 45 60 90 120 135 150 180 radians /2  degrees 210 225 240
Copy ,complete and learn the following table changing degrees to radians
degrees 30 45 60 90
120
135
150
180
radians
/2 
degrees
210
225
240
270
300
315
330
360
radians

7. degrees 30 45 60 90 120 135 150 180 radians /6 /4 /3 /2 2/3 3/430 45 60 90
120
135
150
180
radians
/6
/4
/3
/2
2/3
3/4
5/6 
degrees
210
225
240
270
300
315
330
360
radians
7/6
5/4
4/3
3 /2
5/3
7/4
11/6
2 

8. Changing degrees which are not multiples of 30,45 and 60to radians
Change 70º to radians
180 º =  radians
Start with
Use direct proportion
radians
This is the exact value
Using a calculator does not give the exact value
70º = radians`

9. Change the following angles in degrees to radians give angle as a fraction of  and as a number to 3 dp
degrees
number
fraction
1/9
0.349
20 º
40 º
80 º
140 º
200 º
320 º
144 º
280 º
0.698
2/9
1.396
4/9
2.443
7/9
3.840
11/9
5.59
16/9
2.513
4/5
14/9
4.887

10. Convert radians to degrees
3/2
= 3 X /2
= 3 X 90º
= 270º
7/6
= 7 X /6
= 210º
= 7 X 30º
Other radians measures
 Rads = 180º
Using the calculator
343.8 º
6 rads
= 6/ X 180 º

11. Change the following radian measures to degrees
= 565º
7/X 180
7 rads
5/4 rads
6.28 rads
14 rads
7/12 rads
4.2 rads
1.4 rads
5/3 rads
= 5 X 45º
= 225º
5 X /4
6.28/ x180
= 360º
14/ x 180
= 802º
= 105º
= 7 X 15
7 X /12
4.2/ X 180
= 240.6º
1.4/ X180
= 97º
5 X /3
= 5 X 60
= 300º

12. Sin cos and tan of radian measures
y = sin x 1
90º
360º
degrees
180º
270º 
3/2
Radians 2
/2 -1
When you draw graphs use radians on the x axis
Ex 1 p 45 q 4

13. y = cos x 1
3/2 2
/2  -1

14. Trigonometry 3. Exact Values
Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values
Consider
60º 2 2
60º
30º 3 1
To provide values for 30º and 60º

15. rads /6 /4 /3 /2 x 0º 30º 45º 60º 90º Sin xº Cos xº Tan xº  3 2
/6
/4
/3
/2 x 0º
30º
45º
60º
90º
Sin xº
Cos xº
Tan xº  3 2 1 3 2 1 1 3 3

16. For 45º consider this square/ triangle2 1 1
45º 1 1

17. rads /6 /4 /3 /2 x 0º 30º 45º 60º 90º Sin xº Cos xº Tan xº  3 2
/6
/4
/3
/2 x 0º
30º
45º
60º
90º
Sin xº
Cos xº
Tan xº  3 2
1 2 1 3 2
1 2 1 1 3 1 3

18. The exact value of sin /3
= 3/2
= 0
The exact value of cos /2
We can also express these trig ratios to 2 decimal places using the calculator but remember to change to radian mode
= 0
Cos /2
= cos (/2)
Or cos 1.57
= 0
Ex 1 p 45 q 6,7 8