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

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.

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

Comparing radians and degrees B = r O r = 1 radian =  radians = 180 º

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 1/2 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

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

degrees 30 45 60 90 120 135 150 180 radians /6 /4 /3 /2 2/3 3/4 30 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 

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 = 1.22 radians`

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

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

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

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

y = cos x 1 3/2 2 /2  -1

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

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 ½ 1 3 2 1 ½ 1 3 3

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

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 ½ 1 3 2 1 2 1 ½ 1 3 1 3

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