positive angle the angle measured in an anticlockwise direction. negative angle the angle measured...

POSITIVE ANGLE

The angle measured in an anticlockwise

direction.

NEGATIVE ANGLE

The angle measured in a clockwise

direction.y

x

y

x

TRIGONOMETRIC FUNCTION

y

x

1st

QUADRANT

2nd QUADRANT

3rd QUADRANT 4th QUADRANT

SIN, COS, TAN

ADDSUGAR

TO COFFEE

sin, cosine and tangent have

positive values

cosine has positive value

tangent has positive value

sin has positive value

y

x

REFERENCE ANGLE

1st

QUADRANT2nd QUADRANT

3rd QUADRANT 4th QUADRANT

EXAMPLE Calculate the followings:

(a) sin 120

(b) cosine -120

y

x

(a)

y

x

sin 120 = sin ( 180 120 )

= sin 60 cosine (120) = cosine ( 180 ( 120) )

= cosine ( 60)

= cosine 60 = 0.8660

= 0.5

(b)

EXAMPLE Calculate the followings:

(a) sin 845

(b) tan ( -860)

y

x

(a)

sin 845 = sin ( 845 720 )

= sin 125

= 0.8192

= sin (180 125)= sin 55

y

x

(b)

tan ( 860) = tan ( 860 ( 720 ))

= tan ( 140)

= 0.8391

= tan (180 140)= tan ( 40)

DEFINITION OF SINE COSINE AND TANGENT

y

x

Pm,n

1-1

1

-1

0

n

m

1

sin =

hypotenuse

sideopposite

1

n

coordinatey

cos =hypotenuse

sideadjacent

1

m

coordinatex

tan = sideadjacent

sideopposite

m

n

coordinatex

coordinatey

cos

sin

EXAMPLE

P-0.75,-0.9

1-1

1

-1

0

1

y

x

Q-0.8,0.5

Given the points P (-0.75, -0.9) and Q (-0.8, 0.5) on a unit circle as shown in the diagram. Find the values of

(a) cos

(b) sin

(c) tan

(d) tan

(e) cos

(a) cos = - 0.8

(b) sin = - 0.9

(c) tan = 1.2

(d) tan = - 0.625

(e) cos = - 0.75

Do the exercises in SP 2

DEFINITIONS OF SECANT, COSECANT AND COTANGENT

sin

1 cosec

cos

1 sec

tan

1 cot

cos sin

1 cot

sin

cos

The signs of cot , cosec and sec follow the signs of tan , sin and cos in the respective quadrant.

EXAMPLE

1-1

1

-1

0

y

x

Q-0.78,0.6

Given the points Q (-0.78, 0.6) is on a unit circle as shown in the diagram. Find the values of

(a) cosec 143.13

(b) sec 143.13

(c) cot 143.13

(d) tan 143.13

(e) cos 143.13

143.13

cosec 143.13

(a) = 143.13 sin

1

= 143.13-180 sin

1

6.873 sin

1=

= 60

1

.=

6671.

Determine the value of the

reference angle

sec 143.13(b) = 143.13 cos

1

= 143.13-180 cos

1

6.873 cos

1=

= 780

1

.=

2821.

cot 143.13(c) = 143.13 tan

1

= 143.13-180 tan

1

6.873 tan

1=

= 60

780

.

.=

31.

TRY (d) and (e) Do the exercises in SP 3

EXAMPLE Determine the value for each of the following trigonometric functions.

(a) cosec 140 (b) cot -⅔

y

x

140

cosec 140

(a)

= 140 sin

1

= 140-180 sin

1

40 sin

1=

= 64280

1

.=

5561.

y

x

-⅔

cot -⅔

(b)

= 120 - tan

1

= 120-180 tan

1

60 tan

1=

= 73211

1

.=

57730.

Do the exercises in SP 4

2

1

2

1

3

60

30360 tan

2

160 cos

2

360 sin

2

130 sin

2

330 cos

3

130 tan

SPECIAL ANGLES 30, 45 , 60

2

1

1

45

2

2

145 sin

2

145 cos

145 tan

EXAMPLE Without using a calculator, determine the value for each of the following trigonometric functions.

(a) cot 240 (b) tan -225

y

x

(a)

240

1240

tancot

240

)(tan

180240

1

60

1

tan

3

1

y

x

(b)

225

1225

cottan

-225

)(tan

180225

1

45

1

tan

1

Do the exercises in SP 5

SOLVING TRIGONOMETRIC EQUATIONS

EXAMPLE

Solve the following trigonometric equations for 0 360.

(a) sin - 0.6532 (b) cos 2 = - 0.6824

(a)

sin - 0.6532-ve shows the

quadrant

sin-1 0.6532

40.78 // 40 47’reference

angle

y

x40.78 40.78

180 40.78 // 180 40 47’ ,

360 - 40.78 // 360 - 40 47’ .

220.78 // 220 47’ ,

319.22 // 319 13’ .

(b)

cos 2 = - 0.6824

0 2 720

2 cos-1

0.68242 46.97 // 46 59’

reference angle

y

x

46.97

46.97

2 180 - 46.97 , 180 + 46.97 ,

2 133.03 ,

226.97 ,

360 + 133.03,

360 + 226.97

586.97 . 493.03 ,

66.52, 113.49, 246.52, 293.49 .

12 ½x+12 192

EXAMPLE

Solve 4 cos (½x + 12) + 1 3.626 for 0 x 360.

4 cos (½x + 12) + 1 3.626

SOLUTION

4 cos (½x + 12) 3.626 - 1

cos (½x + 12) 4

6262.

½x + 12 cos-1 0.6565

½x + 12

½x 48.97 - 12

½x 36.97

x 2 ( 36.97 )

x 73.94

48.97

Do the exercises in SP 6

SOLUTION FOR SP 7, N0 1

(a)

y

x5

13 12

0 180

(b)

(i) sin = 13

12

(ii)

tan = 5

12

(iii)

cosec = sin

1

13121

12

13

(iv)

sec = cos

1

1351

5

13

Do the exercises in SP 7

90

EXAMPLE

Solve 6 tan x – 3 cot x = 7 for 0 x 360.

SOLUTION

6 tan x – 3 cot x = 7

6 tan x – 3

xtan

1- 7 = 0

6 tan2 x – 3- 7tan x = 0

6 tan2 x – 7tan x -3 = 0

03213 xx tantan

013 xtan 032 xtan

3

1xtan 2

3xtan

5734157161 .,.x 312363156 .,.x

5734157161312363156 .,.,.,.x

EXAMPLE

Solve 2 cos x=sec x for 0 x 360.

SOLUTION

02 xx seccos

01

2 x

xcos

cos

012 2 xcos

2

1cos x

7071.0cos x 7071.0cos x

315,45x 225,135x

315,225,135,45x