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The trigonometric Ratios, Functions,
Equations and Identity
Trigonometric Ratios
Someone wants to measure a flag pole by
clinometers

Adaptif
Trigonometric Ratios
Learning Experience
 A student of building construction technique
wants to make a house roof frame, and the
size as follows. Then the length X is …
2m
xm
3
4
m
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Trigonometric Ratios
Learning Experience
 The top of tower from A with elevation angle is
300 and seen from B with elevation angle 450 like
in the picture. If the distance A and B is 20
meters, then what is the height of that tower?
300
A
450
B
20 m
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What is going on?
Trigonometric Ratios
If there are several teachers give task for
the students, as follows:
“A right triangle ABC has sides AC=4, BC=6 and
AB=8. Then determine the A side.”
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Trigonometric
Ratios
A Glance
???
 There isn’t any strange in the exercise that is
given by the teacher, is there?
 The students try to calculate the side A by
previously calculate Sinus A value
 The teachers don’t feel guilty
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Trigonometric Ratios
How many meters the height of stair?
3M
4M
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Trigonometric Ratios
Which is the longest circumference of the shape ?
1)
3)
2)
4)
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Trigonometric Ratios
The Scope
1. Trigonometric Ratios of side
2. Specific Angles ( special )
3. Trigonometric Formula
4. Cartesius Coordinat and Pole
5. Sine, Cosine Rules and Triangle Areas
6. Trigonometric Identity
7. Trigonometric Equation
Adaptif
Trigonometric Ratios
SINE
IS A COMPARISON
BETWEEN SIDE OF FRONT
ANGLE AND HYPOTENUSE
OF A RIGHT SIDE TRIANGLE
C
AC
Sin AOC =
OC
0
A
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Trigonometric Ratios
 Cosine is a comparison value between side of angle side and
hypotenuse of right side triangle
C
Cos AOB =
O
OA
OC
A
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Trigonometric Ratios
 Tangent is ratio value between side of angle front
and side of angle side
C
AC
Tan AOC = OA
O
A
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Trigonometric Ratios
Angle in the Base Position
C
Y
B
C
θ
θ
A
Angle θ isn’t in the
base position
B
A
X
Angle θ is in the base position
Side AB is beginning side from angle θ
Side AC is limitation of side from angle θ
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TRIGONOMETRIC RATIOS
SINE CONCEPT
BB' CC' DD' EE'



 ...
AB AC AD AE
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TRIGONOMETRIC RATIOS
COSINE CONCEPT
AB' AC' AD' AE'



 ...
AB AC AD AE
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TRIGONOMETRIC RATIOS
TANGENT CONCEPT
BB' CC' DD' EE'



 ...
AB' AC' AD' AE'
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TRIGONOMETRIC RATIOS
Given triangle ABC, the angle is at C. The lenght AB
side = 10cm, BC side = 5cm.
Cos A value and tan A orderly are....
10
A
? gotten 5V3
B
5
C
Maka diperoleh : sin A = ½
Jadi : cos A = ½ V3
tan A = 1/3 V3
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TRIGONOMETRIC RATIOS
The expanded exercise
By measuring the length of BC
stairs, and measuring the size of
ABC angle, and using the sine
concept, then the students are
given task to determine the
height of second floor from
base floor.
C
Stair
A
B
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TRIGONOMETRIC RATIOS
C
By measuring the size of BAC angle and
AB distance, and also using cosine
concept, then the students can determine
the length of the pole rope AC that must
be changed!
Pole rope
Pole
A
B
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TRIGONOMETRIC RATIOS
Specific Angles
C
A
D
Equilateral ABC
The sides = 2a
B
S
R
P
Q
A square PQRS
The sides = 2a
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TRIGONOMETRIC RATIOS
By using the picture above, determine
the ratios value:

0o
300
450
600
900
….
….
….
….
….
….
….
….
….
….
tg
….
….
….
….
….
ctg
sec
cos ec
….
….
….
….
….
….
….
….
….
….
….
….
….
….
….
sin 
cos
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TRIGONOMETRIC
SpecificRATIOS
Angles
45o
V2
45o
1
90o
1
sin 45o = ½ V2
cos 45o = ½ V2
tan 45o = 1
sin 30o = ½
cos 30o = ½ V3
30o
2
tan 30o = 1/3 V3
V3
sin 60o = ½V3
90o
60o
1
cos 60o = ½
tan 60o = V3
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TRIGONOMETRIC RATIOS
TROGONOMETRIC FORMULAS
A. Relation/Base Formula of Trigonometric Function
1. a. Opposite Relation:
1
1
1
cot α = tan 
sec α = cos 
csc α = sinα
b. Division Relation:
sinα
tan α = cosα
cos α
cot α =
sin α
c. “Pythagoras” Relation:
sin2α + cos2α = 1 (and its variety)
tan2α + 1 = sec2α
1 + cot2α = csc2α
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TRIGONOMETRIC
RATIOS
Related Angles
2. Trigonometric Function of related angles
a. sin(90 – α)o = cos αo
cos(90 – α)o = sin αo
tan(90 – α)o = cot αo
cot(90 – α)o = tan αo
sec(90 – α)o = csc αo
csc(90 – α)o = sec αo
b.
sin(180 – α)o = sin α0
cos(180 – α)o = –cos α0
tan(180 – α)o = –tan α0
sin(180 + α)o = –sin αo
cos(180 + α)o = –cos αo
tan(180 + α)o = tan αo
c.
sin(360 – α)o = –sin α0
cos(360 – α)o = cos α0
tan(360 – α)o = –tan α0
sin(–αo) = –sin αo
cos(–αo) = cos αo
tan(–αo) = –tan αo
Value
”+”
Sin
Tan
All
Cos
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TRIGONOMETRIC RATIOS
Specific Things
1. If αo + βo + γo = 180o , then:
sin(α + β)o = sin(180 – γ)o = sin γo
cos(α + β)o = cos(180 – γ)o = –cos γo
sin ½ (α + β)o = sin(90 – ½ γ)o = cos ½ γo
cos ½ (α + β)o = cos (90 – ½ γ)o = sin ½ γo
2. If αo + βo + o = 270o, then:
sin(α + β)o = sin(270 – )o = –cos o
cos(α + β)o = cos(270 – )o = –sin o
Adaptif
Cartesians Coordinate and Pole
Y
P( x,y )
x
Y

x
r
y
x
o
Cartesians
PoleCoordinate
Coordinate to Cartesians
x = r cos a
Y = r sin a
P( r,  )

y

O
X
Pole Coordinate
Cartesians Coordinate to Pole
r2 = x2 + y2
y
tan α =
x
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Trigonometric Formula in Triangles
1. Sine Rules (formula) in ABC Triangles:
a
b
c


sin  sin  sin 
2. Cosine Rules (formula):
a2 = b2 + c2 – 2bc cos α
b2 = a2 + c2 – 2ac cos β or
c2 = a2 + b2 – 2ab cos γ
b2  c 2  a 2
cos α =
2bc
c 2  a 2  b2
cos β =
2ca
cos γ = a2  b2  c2
2ab
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Trigonometric Formula in Triangles
From a quay, boat A travels with 10 knot speed (mil/hour)
to the 160o and boat B to the 220o with 16 knot speed. So
what is the distance of two boats two hours later?
U
AB2 = 202 + 322 – 2. 20 . 32 . cos 60o
= 400 + 1024 – 640
220o
O
160o
= 784
60o
20
32
B
AB = 28
A
The distance between two boats
is 28 mil
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Trigonometric Formula in Triangles
C
20
37
51
A
Find tan A value and sin B?
cos A =
so cos B =
cos B =
so sin A =
B
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TRIGONOMETRIC FUNCTION FORMULAS OF TWO
ANGLES
The Difference Formula
1. Addition Formula
 sin(α – β) = sin α cos β – cos α sin β
 sin(α + β) = sin α cos β + cos α sin
β
 cos(α – β) = cos α cos β + sin α sin β
 cos(α + β) = cos α cos β – sin α sin
β
tan(    ) 
tan   tan 
1  tan  tan 
tan (   ) 
A half Angle Formula
2. Double Angle Formula


Sin 2α = 2 sin α cos α
2
2
Cos 2α = cos
2 tanα – sin α
tan 2 
1  tan 2 
tan   tan 
1  tan  tan 


2 sin2 ½ α = 1 - cos α
2 cos2 2½ α = 11+cos
cos
α
1
tan 2  
1  cos 
sin 
1
tan 2  
1  cos 
Adaptif
TRIGONOMETRIC FUNCTION FORMULA OF
TWO ANGLES
3. Triple angles formula
Sin 3α = 3 sin α – 4 sin3 α
Cos 3α = 4cos3α – 3 cos α
3 tan   tan 3 
tan 3 
1  3 tan 2 
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The difference, Addition Formula and
Division Result of Sine/cosine Function
1. Multiplication Result of sine and cosine




2 sin α cos β = sin(α + β) + sin(α – β)
2 cos α sin β = sin(α + β) – sin(α – β)
2 cos α cos β = cos(α + β) – cos(α – β)
–2 sin α sin β = cos(α + β) – cos(α – β)
or 2 sin α sin β = cos(α – β) – cos(α + β)
2. Addition and Difference of Sine/Cosine Function




sin A + sin B = 2 sin ½ (A + B) cos ½ (A – B)
sin A – sin B = 2 cos ½ (A + B) sin ½ (A – B)
cos A + cos B = 2 cos ½ (A + B) cos ½ (A – B)
cos A – cos B = –2 sin ½ (A + B) sin ½ (A – B)
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TRIGONOMETRIC IDENTITY
 Identity is an open sentence which has true
value for every its variable value substitution
for example : sin2α + cos2α = 1

Prove !
sin x
1  cos x
2 csc x 

1  cos x
sin x
sec4 – sec2 = tan4 + tan2
Adaptif
TRIGONOMETRIC IDENTITY
Prove:
sin x
1  cos x

1  cos x
sin x
sin 2 x  (1  cos x ) 2

(1  cos x ) sin x
sin 2 x  1  2 cos x  cos 2 x

(1  cos x ) sin x
2  2 cos x

(1  cos x ) sin x
2

sin x
 2 csc x  right space (proven)
Adaptif
TRIGONOMETRIC IDENTITY
Prove:
Alternative I From left space
 Left space:
sec4 – sec2
Alternative II From right space
Right space:
tan 4 + tan 2
= sec2(sec2 – 1)
= tan 2(tan 2 + 1)
= sec 2 x tan 2
= (sec 2 – 1) sec 2
= (1 + tan 2) x tan 2
= sec 4 – sec 2
= tan 2 + tan 4
= left space (proven)
= tan 4 + tan 2
= right space (proven)
Adaptif
Simple Trigonometric Equation
Formula I :
1). If sin x  sin 
then: x   + k. 360 or
x  (180  ) + k. 360 , k  B
2). If cos x  cos 
then : x   + k. 360 or
x    + k. 360, k  B
3). If tan x  tan 
then : x   + k. 180 k  B
Adaptif
TRIGONOMETRIC EQUATION
Formula II : At the same condition is equal to zero
1). If sin x  0
then: x  k.180 , k  B
2). If cos x  0
then: x  90 + k.180 , k  B
3). If tan x  0
then: x  k.180 , k  B
Adaptif
TRIGONOMETRIC EQUATION
Formula III : Equation contains negative value
1). If sin x  - sin   sin (-)
then: x  -  + k. 360 or
x  (180 + ) + k. 360 , k  B
2). If cos x  - cos   cos (180 +  )
then: x  180 +  + k. 360 or
x  - 180 -  + k. 360 , k  B
3). If tan x  - tan   tan (-)
then: x  -  + k. 180 , k  B
Adaptif
TRIGONOMETRIC EQUATION
Example
 Determine the soluton set of the trigonometric equation below:
For 0 ≤ x < 360:
 a) sin x0 = sin 400
b) cos 2x0 =
Answer:
1
2
 a) sin x0 = sin 400  x = 40 + k.360 or x = (180 – 40) + k.360

For k = 0 → x = 40
k = 0 → k = 140
 So the solution set is {40, 140}
 is {30, 150, 210, 330}
Adaptif
TRIGONOMETRIC EQUATION
1
=
2
cos 2x 0 = cos 60
 b) cos 2x
0
0
then 2x = 60 + k.360 or 2x = -60 + k.360
x = 30 + k.1 80
 for

k = 0 → x = 30
for
k = 1 → x = 150
x = -30 + k.180
k = 1 → x = 2100
k = 2 → x = 330
 So the solution set is
{30, 150, 210, 330}
Adaptif
TRIGONOMETRIC EQUATION
Exercise :
1.Given ABC triangle, AC =25 cm, BC=40 cm,
and the length of high line from C, is CD=24 cm.
Cos A and tan B value orderly is....
2.
Two boats leave a quay at the same time. The
first boat sails in the bearing of 030° with 8
km/hour, while the second boat sails in the
bearing of 090° with 10 km/hour. The distance
of two boats after sailing for 3 hours is ... km
Adaptif
Application to the competence program
A roof frame of a building is like the next
picture:
 Calculate the length of AB
B
A
2,20 m
35,30
28,50
10,30 m
The length of AB is 3,14 m
Adaptif
Application to the competence program
A
See the picture:
18 cm
400
B
950
700
C
a) Calculate the
distance of AB
b) Calculate the
distance of BC
a) The distance of AB
= 12,6 cm
b) The distance of BC
= 21,97 cm
Adaptif