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TRIGONOMETRY
PYTHAGORAS’ THEOREM
Square on Hypotenuse
=
+
Square on Leg 1
hyp  leg1  leg 2
2
Square on Leg 2
2
2
DEdwards
TRIGONOMETRIC FUNCTIONS
hyp
opp

adj

Sine
sin  

opp
hyp
 opp 

  sin 
 hyp 
1
Cosine
cos  
adj
hyp
 adj 

  cos 
hyp


1

Tangent
tan  
opp
adj
 opp 

 adj 
  tan 1 
TRIGONOMETRIC IDENTITIES
opp
hyp

adj
Prove:
 tan
. 
sin 
cos 
 .sin 2   cos 2   1
TRIGONOMETRIC FUNCTIONS
Special Angles
proof
proof
0°
30°
Sin θ
?
0
?2
Cos θ
?1
?
0
θ
Tan θ
60°
90°
?2
?3
?1
?2
3
1
?2
?2
1
?
0
1
?3
?1
1
45°
proof
1
2
?3
?
undefined
DEdwards
TRIGONOMETRIC FUNCTIONS
60°
30°
2
3
60°
opp
hyp
 12

Sin 30° =

Cos 30° = hyp
adj


Tan 30° =
opp
adj

1
2
2 2  12
3
60°
2
opp
hyp

Sin 60° =
3
2

Cos 60° =
adj
hyp
1
3

Tan 60° opp
=

adj

3
2
 12
3
1
 3
table
TRIGONOMETRIC FUNCTIONS
45°
2
1
2
1

opp
Sin 45° = hyp

1
2
adj

Cos 45° = hyp

1
2

Tan 45° = opp
adj
12  12
 11  1
table
GRAPHS OF TRIGONOMETRIC FUNCTIONS
Trigonometric Graphs are PERIODIC i.e. they
repeat themselves after a “cycle” is complete
GRAPHS OF TRIGONOMETRIC FUNCTIONS
max(sin x)  1
sin(x)
1.0
y-intercept = 0
0.8
0.6
0.4
0.2
0.0
-360
-270
-180
-90
-0.2
sin(x)
0
90
180
270
360
-0.4
-0.6
-0.8
-1.0
x-intercepts =
0 °, ±180°, ±360°
min(sin x)  1
GRAPHS OF TRIGONOMETRIC FUNCTIONS
max(cos x)  1
cos(x)
1.0
y-intercept = 1
0.8
0.6
0.4
0.2
0.0
-360
-270
-180
-90
-0.2
cos(x)
0
90
180
270
360
-0.4
-0.6
x-intercepts =
-0.8 ±90°, ±270°
-1.0
min(cos x)  1
GRAPHS OF TRIGONOMETRIC FUNCTIONS
The cosine curve is the sine curve
translated 90° left
1.0
0.8
0.6
0.4
0.2
0.0
-360
-270
-180
-90
-0.2
-0.4
-0.6
-0.8
-1.0
0
90
180
270
360
sin(x)
cos(x)
TRIGONOMETRIC RELATIONSHIPS

Complementary Relationships
 Sin
x = Cos ( 90 - x)
 Cos
 Tan
x = Sin ( 90 - x)
x=
1
Tan(90  x)
TRIGONOMETRIC RELATIONSHIPS

Supplementary Relationships
 Sin
x = Sin ( 180 - x)
 Cos
x = - Cos ( 180 - x)
 Tan
x = - Tan ( 180 - x)
GRAPHS OF TRIGONOMETRIC FUNCTIONS
tan x  undefined , when x  90,270
tan(x)
1.0
The function has
ASYMPTOTES at these points
0.8
0.6
0.4
0.2
0.0
-360
-270
-180
-90
-0.2
-0.4
-0.6
-0.8
-1.0
tan(x)
0
90
180
270
360
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE
Circle on the Cartesian plane with
radius of 1 unit
sin(x)
(0,1)
0.8
0.6
0.4
0.2
(-1,0)
-1
(1,0)sin(x)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE
sin(x)
(0,1)
Other side forming
desired angle
0.8
0.6
0.4
0.2
(-1,0)
-1
(1,0)sin(x)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
0.4
0.6
0.8
1
-0.4
Positive side of x axis
-0.6
-0.8
-1.0
(0,-1)
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE:
1st Quadrant
sin(x)
(0,1)
  
0.8
0.6
0.4
0.2
(-1,0)
-1
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
Theta θ :
Angle between
Terminal side & Initial side
Reference Anglesin(x)
α:
Acute Angle(1,0)
between
Terminal Side & x-axis
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE:
2nd Quadrant
sin(x)
(0,1)
  180  
0.8
0.6
(-1,0)
-1
0.4
Theta , θ
0.2
Reference Anglesin(x)
α
(1,0)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE:
3rd Quadrant
sin(x)
(0,1)
  180  
0.8
0.6
(-1,0)
-1
0.4
Theta , θ
0.2
Reference Anglesin(x)
α
(1,0)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE:
4th Quadrant
sin(x)
(0,1)
  360  
0.8
0.6
0.4
Theta , θ
0.2
(-1,0)
-1
Reference Anglesin(x)
α
(1,0)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
THE UNIT CIRCLE
1.0
sin(x)
(0,1)
Coordinates of points on the unit circle show
(cos θ , sin θ)
0.8
When the terminal side is drawn through that
point
0.6
0.4
0.2
(-1,0)
-1
(1,0)sin(x)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
THE UNIT CIRCLE
sin(x)
(0,1)
(cos θ , sin θ)
0.8
0.6
->
P(a
x , b)
b
0.4
0.2
-1
-0.8
-0.6
-0.4
0.0 O
-0.2
0
-0.2
-0.4
-0.6
<-
0.2
0.4
->
0.6
0.8
X
1
In the triangle POX
𝑎𝑑𝑗
Cos θ °= ℎ𝑦𝑝
= 𝑎1 = 𝑎
𝑜𝑝𝑝
sin θ °= ℎ𝑦𝑝
= 𝑏1 = 𝑏
-0.8
-1.0
a
(1,0)sin(x)
<-
(-1,0)
θ°
(0,-1)
Hence any point (a,b) on the circle shows:
(cos θ , sin θ)
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
THE UNIT CIRCLE
sin(x)
(0,1)
Coordinates of points on the unit circle show
1.0 x
(cos θ , sin θ)
0.8
When the terminal side is drawn through that
point
0.6
θ=90 °
cos90 °=0
sin90 ° =1
0.4
0.2
(-1,0)
-1
(1,0)sin(x)
Hence point on circle is
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
(0,1)
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
THE UNIT CIRCLE
1.0
sin(x)
(0,1)
Coordinates of points on the unit circle show
(cos θ , sin θ)
When the terminal side is drawn through that
point
0.8
0.6
θ=180 °
Cos180° = -1
Sin 180° = 0
0.4
0.2
(-1,0)
x
-1
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
(-1,0)
0.2
-0.4
-0.6
-0.8
-1.0
(1,0)sin(x)
Hence point on circle is
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
THE UNIT CIRCLE
1.0
sin(x)
(0,1)
Coordinates of points on the unit circle show
(cos θ , sin θ)
When the terminal side is drawn through that
point
0.8
0.6
0.4
0.2
(-1,0)
(1,0)sin(x)
Hence point on circle is
0.0
-1
θ=270 °
Cos 270° = 0
Sin 270° = -1
-0.8
-0.6
-0.4
-0.2
0
-0.2
(0, -1)
0.2
0.4
0.6
0.8
1
-0.4
-0.6
-0.8
-1.0 x(0,-1)
DEdwards
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
THE UNIT CIRCLE
θ=30 °
sin(x)
(0,1)
Cos 30 °= 0.866=0.9(1dp) (cos θ , sin θ)
1.0
sin30 ° = 0.5
0.8
Hence point on circle is
(0.9 , 0.5)
0.6
x (0.9 , 0.5)
0.4
0.2
(-1,0)
-1
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(1,0)sin(x)
θ=30 °
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
1.0
sin(x)
(0,1)
THE
QUADRANT
0 <θ ≤ 90
For all points (x,y)
x is positive & y is positive
So, all points (cos θ,sin θ )
Cos θ :positive
Sin θ : positive
Tan θ = sin θ /cos θ=positive
0.8
0.6
0.4
0.2
(-1,0)
-1
1st
ll are Positive
sin(x)
0.0
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
INE (only) is Positive
2nd
THE
QUADRANT
90 ° <θ ≤ 180°
For all points (x,y)
x is negative & y is positive
So, all points (cos θ,sin θ )
Cos θ :negative
Sin θ : positive
Tan θ = sin θ /cos θ=negative
1.0
sin(x)
(0,1)
ll are Positive
0.8
0.6
0.4
0.2
sin(x)
0.0
-1
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
-0.4
-0.6
-0.8
-1.0
(0,-1)
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
INE (only) is Positive
1.0
sin(x)
(0,1)
ll are Positive
0.8
0.6
0.4
0.2
sin(x)
0.0
-1 THE-0.8
-0.6
-0.4
-0.2
0
0.2
3rd QUADRANT
-0.2
180 ° <θ ≤ 270°
For all points (x,y)
-0.4
x is negative & y is negative
So, all points (cos θ,sin θ )
-0.6
Cos θ :negative
-0.8
Sin θ : negative
Tan θ = sin θ /cos θ=positive
-1.0 (0,-1)
AN (only) is Positive
0.4
0.6
0.8
1
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
INE (only) is Positive
1.0
sin(x)
(0,1)
ll are Positive
0.8
0.6
0.4
0.2
0.0
-1
-0.8
-0.6
AN (only) is Positive
-0.4
sin(x)
th
-0.2
0 THE 4
0.2QUADRANT
0.4
0.6
0.8
1
-0.2 270 ° <θ ≤ 360°
For all points (x,y)
-0.4 x is positive & y is negative
-0.6 So, all points (cos θ,sin θ )
Cos θ :positive
-0.8 Sin θ : negative
Tan θ = sin θ /cos θ=negative
(0,-1)
-1.0
OS (only) is Positive
TRIGONOMETRIC FUNCTIONS ON THE CARTESIAN PLANE
INE (only) is Positive
1.0
sin(x)
(0,1)
ll are Positive
0.8
0.6
0.4
0.2
sin(x)
0.0
-1
-0.8
-0.6
-0.4
-0.2
0
-0.2
0.2
0.4
0.6
0.8
1
-0.4
-0.6
-0.8
AN (only) is Positive
-1.0
(0,-1)
OS (only) is Positive
TRIGONOMETRIC RELATIONSHIPS

“CAST” Relationships

Sin x = Sin (180 - x) = -Sin (180 + x) = Sin ( 360 - x)

Cos x = -Cos (180 - x) = -Cos (180+ x) = Cos (360 - x)

Tan x = -Tan (180 - x) = Tan (180+ x) = -Tan(360 - x)