Trigonometry Functions slides

1. Trigonometry

2. Trigonometry (from Greek trigōnon
"triangle" + metron "measure")[1] is a
branch of mathematics that deals with
triangles,
Trigonometry deals with relationships
between the sides and the angles of
triangles and with the trigonometric
functions, which describe those
relationships.

3. The Canadarm2 robotic manipulator on the
International Space Station is operated by controlling
the angles of its joints. Calculating the final position of
the astronaut at the end of the arm requires repeated
use of the trigonometric functions of those angles.

5. ©CarolynC.Wheater,2000
5

7. INTRODUCTION.

8. -1
-1
1
1
y
x
o
III
III IV
Identifying quadrants and angles in the
unit circle.
The Unit Circle is a circle of radius 1 unit with its
centre (0,0), on the Cartesian plane.The x-axis
and the y-axis divide the circle into four equal
quadrants.

9. -1
-1
1
1
y
x
o
rad
2
/900 
rad
2
3
/2700 
rad/1800
rad2/3600
rad0/00
III
III IV
00
00
00
00
360270,
270180,
18090,
900,








IVQuadrant
IIIQuadrant
IIQuadrant
IQuadrant

10. Understand and use
the concept of
positive and negative angles.

11. An angle is measured from an
initial side which is the positive
x-axis to its terminal side.

12. There are 2 radians in a full rotation -
- once around the circle
There are 360° in a full rotation
To convert from degrees to radians or
radians to degrees, use the proportion
2360
deg
0
radiansrees


13. y
x

o
Terminal side
Initial side
When the angle is measured in an
anticlockwise direction from the initial side
to the terminal side, it is a positive angle.
Anti-clockwise
direction

14. y
x

o
Terminal side
Initial side
When the angle is measured in an
clockwise direction from the initial side to
the terminal side, it is a negative angle.
clockwise direction

15. Represent each of the following angles
in a Cartesian plane.Hence, state which
quadrant the angle is in.
0
45)(a
0
200)( b

3
1
)( c

3
2
)( d

16. 0
45
0
200
Quadrant I
Quadrant II

3
1


3
2

(a)
(d)
(c)
(b)
Quadrant IV
Quadrant III

17. Represent each of the following
angles in a Cartesian plane.Hence,
state which quadrant the angle is in.
0
400)(a
0
540)( b

4
15
)(c

5
11
)( d

18. 0
400
0
540
Quadrant I
Quadrant III

5
11

(a)
(d)
(c)
(b)

4
7
Quadrant IV
Quadrant IV
000
40360400 
)180(360540 000

 2
4
7
4
15

)2(
5
1
5
11
 

20. The Unit Circle
Imagine a circle on the
coordinate plane, with
its center at the origin,
and a radius of 1.
Let P, ( x, y) is a point
on the circle somewhere
in quadrant I.
-
1
-
1
1
1 P

21. The Unit Circle
Creates a right triangle
with hypotenuse of 1
from point P
x
y
P(x, y)
1

22. The Unit Circle
x
y
1
 is the
angle of
rotation
From the trigonometric
ratios, this triangle gives
x
x

1
)cos(
x
y
)tan(
)sin,(cos P
y
y

1
)sin(

23. Reciprocal of Trigonometric
Functions
x
y
1
 is the
angle of
rotation
x
1
sec
cos
1
 

y
ec
1
cos
sin
1
 

y
x
 

cot
tan
1
HINT: Refer 3rd
letter

24. Reference Angle
Reference angle

Reference angle , is
the acute angle between
the terminal side and the x-
axis.






25. Find the value of each
trigonometric equation:
0
0
0
0
0
0
145tan)(
145cos)(
145sin))((
70tan)(
70cos)(
70sin)()(
iii
ii
ib
iii
ii
ia
0
0
0
0
0
0
314tan)(
314cos)(
314sin))((
210tan)(
210cos)(
210sin))((
iii
ii
id
iii
ii
ic

26. Quadrant I
Angle given,
Reference
angle
For an angle, , in
quadrant I,
In quadrant I,
sin() is positive
cos() is positive
tan() is positive


 
P(x,y)

27. Quadrant II
Angle given,
Reference
angle,
For an angle, , in
quadrant II,

In quadrant II,
sin() is positive
cos() is negative
tan() is negative


P(-x,y)

28. Quadrant III
Angle given,
Reference angle
For an angle, , in
quadrant III,
-
In quadrant III,
sin() is negative
cos() is negative
tan() is positive



P(-x,-y)

29. Quadrant IV
Angle given,
Reference angle
For an angle, , in
quadrant IV,
2
In quadrant IV,
sin() is negative
cos() is positive
tan() is negative



P(x,-y)

30. Add Sugar To Coffee
 Use the phrase “Add Sugar To Coffee” to
remember the signs of the trig functions in
different quadrants.
AddSugar
To Coffee
All functions
are positive
Sine is positive
Tan is positive Cos is positive
AfterSchool
Terus Cabut

31. The relationship between positive angles
and negative angles in trigonometric
functions


)tan()tan(
)cos()cos(
)sin()sin(






Quadrant IV,
cos( )
positive


32. Given that and
Find the values of the following:
7660.0130sin 0
 6428.0130cos 0

0
0
0
0
130cos)(
130sec)(
130cot)(
130tan)(
ecd
c
b
a

33. 8391.0
130sin
130cos
130cot)(
1917.1
130cos
130sin
130tan)(
0
0
0
0
0
0


b
a
3055.1
130sin
1
130cos)(
5557.1
130cos
1
130sec)(
0
0
0
0


ecd
c

34. State the reference angle for each of the
following angles:
radd
c
radb
a


5
11
)(
610)(
3
5
)(
127)(
0
0

35. 0
00
00
0
00
60
300360
300180
3
5
3
5
)(
53
127180)(








b
a
0
000
0
0
0
00
000
36
36360396
396
180
5
11
5
11
)(
70
180250
250360610)(











d
c

36. Find the value of each of the
following trigonometric functions.
)342(cos)(
258sin)(
0
0
b
a
)
3
8
(cos)(
535cot)( 0
ecd
c

37. 9511.018cos
18
342360
)342cos()342cos()(
9781.0
78sin258sin
78
180258
.3258)(
0
0
00
00
00
0
00
0












b
quadrantrdata
1547.1
60sin
1
)480(cos
480
3
8
)(
4286.11535cot
175tan
1
535cot
)2(175
360535)(
0
0
0
0
0
0
0
00







ec
d
quadrantndat
c




38. Complementary Angles
x
y
)90( 0


)90tan(cot
)90(cossec
)90sec(cos
)90cot(tan
)90sin(cos
)90cos(sin
,
0
0
0
0
0
0












ec
ec
anglesarycomplementFor
Complementary angle
)90( 0


39. Trigonometry Values for Special
Angles
1
2
30°
60°
45°
45°
3
2
1
1
000
604530

40.  sin cos tan
rad2,360,0 00
0
0
0
0
rad
2
/900 
rad/1800
rad
2
3
/2700 
1
1
1
1 undefined
undefined
0
0
Trigonometry Values for Special
Angles

41. Find the value of each the following
function without using calculator
0
0
0
0
)330sin()(
315cos)(
300sin)(
135sec)(
d
c
b
a

42. 2
1
60sin)300sin()(
2
1
45cos315cos)(
2
3
60sin300sin)(
2
45cos
1
135cos
1
135sec)(
00
00
00
0
0
0





d
c
b
a

43. Solving Trigonometric
Equations
STEPS
Determine the quadrant the angle lies
based on the sign of value of the trigo.
function.
Find the reference angle using a
scientific calculator
Determine all the possible angle values
based on the given range.

44. Find the value of for each of the
following trigonometric equations for

00
3600 
9238.2cos)(
1135.2tan)(
5478.0cos)(
2344.0sin)(








ecd
c
b
a

45. 00
01
00
0
1
78.236,22.123
78.565478.0cos
32cos)(
44.166,56.13
56.13
2344.0sin
21sin)(















QandQatveb
QandQattvea
00
0
00
0
340,200
20
3420.0sin
9238.2
sin
1
)(
68.244,68.64
68.64
31tan)(














d
QandQattvec

46. Find the values of x for each of the
following trigonometric equations for 00
3600  x
7883.0)152sin()(
3420.03cos)(
5344.02sin)(
0



xc
xb
xa

47. 000000
000000
00
0
0000
0000
00
33.323,67.276,33.203,67.156,33.83,67.36
970,830,610,470,250,1103
108030
70
3420.03cos)(
85.253,15.196,85.73,15.16
7.507,3.392,7.147,3.322
72020
3.32)(









x
x
xRange
xb
x
x
xRange
a


 
0000
00
000
000
0
49.326,52.288,49.146,52.108
97.667,03.592
,97.307,03.232152
73515215
03.52)(




x
x
xRange
c 

48. Solve the following equations for 00
3600  x
xecxc
xxb
xxa
cos2sin3)(
03coscos10)(
sin3tan2)(
2




49. 0000
2
00000
300,13.233,87.126,60
5.0cos,6.0cos
0310
cos)(
360,81.311,180,19.48,0
3
2
cos,0sin
0)cos32(sin
0sincos3sin2
0sin3
cos
sin
2
0sin3tan2)(










x
xx
yy
yxletb
x
xx
xx
xxx
x
x
x
xxa
000
2
2
270,53.160,47.19
1sin,
3
1
sin
0123
01sin2sin3
0
sin
1
2sin3)(





x
xx
yy
xx
x
xc

50. Known as Pythagorean Identities
because they are obtained from
Pythagoras’ theorem.

51. Pythagorean Identities
51
1cossin 22
   22
sec1tan 

52. 1cossin 22
 
 22
sec1tan 
 22
coscot1 ec

53. Prove each of the following
trigonometric identities
xxxd
x
xc
xxxxb
xxxxa
244
2
2222
2
sin21sincos)(
tan1
1
cos)(
cotcoscotcos)(
1cossin2)cos(sin)(






54. x
x
x
x
x
xecx
xx
LHSb
xx
xxxx
xxxx
LHSa
2
2
2
2
2
22
22
22
22
cot
sin
cos
sin
1
cos
)(coscos
)cot1(cos
:)(
cossin21
cossin2cossin
coscossin2sin
:)(














:)( LHSc
x
xx
xxxx
LHSd
x
x
x
x
RHSc
2
22
2222
2
2
sin21
)1(sin)sin1(
)sin)(cossin(cos
:)(
cos
sec
1
sec
1
tan1
1
:)(









55. BA
BA
BA
BABABA
BABABA
tantan1
tantan
)tan(
sinsincoscos)cos(
sincoscossin)sin(







56. A
A
A
A
A
AAA
AAA
2
2
2
22
tan1
tan2
2tan
sin21
1cos2
sincos2cos
cossin22sin







57. 2
tan1
2
tan2
tan
2
sin21
1
2
cos2
2
sin
2
coscos
2
cos
2
sin2sin
2
2
2
22
A
A
A
A
A
AA
A
AA
A







58. Example
©CarolynC.Wheater,2000
58

59. Example
©CarolynC.Wheater,2000
59

60. Example
©CarolynC.Wheater,2000
60

61. Example
61

62. Example
62

63. Example
63

64. Example
64

65. Example
65

66. Example
66

67. Example
67

68. Solve the following equations for 00
3600  x
4tan2sec)(
1sincos2)(
2
2


xxb
xxa

69. 000
2
2
2
2
330,210,90
1sin,
2
1
sin
1sin
01sin
01sinsin22
1sin)sin1(2)(






x
xx
x
x
xx
xxa
0000
00
2
2
315,135,57.251,57.71
45,57.71
1tan,3tan
03tan2tan
04tan2tan1)(





x
xx
xx
xxb


70. Solve the following equations for 00
3600  x
0
00
60cos2sin)(
0cos2cos)(
)30cos()30sin()(



xc
xxb
xxa

71. Given that and
and the angles A and B are in the
first quadrant and third quadrant
respectively, find the value of
tan(A-B)
13
5
sin A
5
3
sin

B

72. If and
express the following in terms
of m and/or n.
0
36cosm 0
44sinn
0
0
8cos)(
80sin)(
b
a

73. Given that ,
find the value of without
using a calculator
00
900,
13
5
cos  
2
sin


75. Graphs of Trigonometric
Functions.
X-axis is in angle measures
(degrees or radians)
Y-axis is in numbers.
Trigonometric functions is a
periodic function

76. sinx + circle
90
o
180o

0o 270
o
1
-1
Graphs of
sine
function:
0 90 180 360270-90-180-270-360
360
o
0o
90
o
180o
270
o

77. Period = 3600
Amplitude =1
Maximum and minimum values =+1
and -1
X-intercepts are
Y-intercept= 0
000
360,180,0

78. General equation
y = a sinb x+c
The value of a affects the
amplitude

79. ksinx
27
0
-
360
9
0
18
0
x
y = f(x)
0 36
0
-90-
180
-
270
1
-1
2
-2
sinx
2sinx
3
-3
3sinx
y = ½sinx
Amplitude =2 Amplitude =1
Amplitude =3
Amplitude =1/2
Period
360o

80. General equation
y = a sin b x+c
The value of b affects the period
of the graph.
b
or
b
period
0
3602


81. sin kx x 4
xx
x x
y = sinx y = sin2x
y = sin3x y = sin ½ x
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o

82. General equation
y = a sin b x+c
The value of c affects the
vertical translation

83. ksinx
270-360 90 180
x
0 360-90-180-270
1
-1
2
-2
3
-3
y =sinx+2
y =sinx
y =sinx-2

84. General equation
y = a sin b (x+d)+c
The value of d affects the
horizontal translation

85. ksinx
270-360 90 180
x
0 360-90-180-270
1
-1
2
-2
3
-3
y =sin(x-2)
y =sinx
y =sin(x+2)

86. sinx + circle
90
o
180o

0o 270
o
1
-1
Graphs of
cosine
function: 0 90 180 270-90-180-270-360 360
360
o
0o
90
o
180o
270
o

87. Period = 3600
Amplitude =1
Maximum and minimum values =+1
and -1
X-intercepts are
Y-intercept= 0
00
270,90

88. General equation
y = a cosb x+c
The value of a affects the
amplitude

89. kcosx
9
0
18
0
x
y =
f(x)
0 27
0
36
0
-90-
180
-
270
-
360
1
-1
2
-2
3
-3
cosx
½cosx
2cosx
3cosx

90. General equation
y = a cos b x+c
The value of b affects the period
of the graph.
b
or
b
period
0
3602


91. cos kx x 4
xx
x x
y =cosx y = cos2x
y = cos3x y = cos ½ x
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o

92. General equation
y = a cos b x+c
The value of c affects the
vertical translation

93. kcosx
90 180
x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
3
-3
Cos x
Cos x+2
Cos x-2

94. General equation
y = a cos b ( x+d)+c
The value of d affects the
horizontal translation

95. kcosx
90 180
x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
3
-3
Cos x
Cos (x-90)0
Cos (x+90)0

96. The graphs of absolute values of
the sine and cosine can be
drawn by reflecting the negative
portion over the x-axis.
xyxy cossin 

97. 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o
y =sinx
xy sin

98. xx
y = sinx
y = -2sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = - sinx
y = 2sinx

99. x
y = cosx
y = 2cosx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = -cosx
y = -2cosx
x

100. General equation
y = a tan b x+c

101. Period = 1800
No Maximum and minimum values
Function undefined at 00
270,90x

102. y = tan

0o 90
o
180
o
270
o
360
o
-90o-180o-270o-360o
450
o
-
450o
x

103. Sketch the graph of each following
trigonometric functions for
xyc
xyb
xya
tan)(
2sin3)(
2cos2)(



20  x

104. The number of solutions to a trigonometric
equation involving non-trigonometric
expressions:

105. Separate the trigo. Expression from
the non-trigo. Expression.
Sketch the graphs of the to
functions on the same axis
Determine the number of points of
intersection =number of solutions.

106. Sketch the graph of
. On the same
axes, sketch the line
.Hence, find the number of
solutions to the equation
202cos3  xforxy

x
y
2

 20022cos3  xforxx