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