Page 95 - WCS - Electrical

Workshop Calculation & Science - Electronics Mechanic
Exercise 1.6.26
Workshop Calculation & Science - Electronics Mechanic
Trigonometry - Trigonometrical ratios
Trigonometry - Trigonometrical ratios
Dependency
1
AC
1
sec
θ 
Dependency


1
AC
1
AB
θ
cos 
θ
AB 

sec
The sides of a triangle bear constant ratios for a given
AB
AB
The sides of a triangle bear constant ratios for a given
AC
definite value of the angle. That is, increase or decrease in
AC
definite value of the angle. That is, increase or decrease in
the length of the sides will not affect the ratio between them
1
1
AB
the length of the sides will not affect the ratio between them
unless the angle is changed. These ratios are trigonometrical
cot

θ 

1
1
AB
unless the angle is changed. These ratios are trigonometrical

BC 
θ

cot
ratios. For the given values of the angle a value of the ratios
BC
tanθ
ratios. For the given values of the angle a value of the ratios
AB
AB
BC
BC
AB
AC
AC
AB
,
,
,
and
,
do not change even when
AC
BC
BC
AB
AC
AB
sideBC
a
BC
AB
AC ,
BC ,
and
AB ,
AC ,
sin
do not change even when
θ 

sideBC
AB
BC
BC
AB
AC
AC
b
sideAC
sin
θ 

the sides AB, BC, AC are increased to AB', BC' and AC' or
b
sideAC
the sides AB, BC, AC are increased to AB', BC' and AC' or
decreased to AB", BC" and AC".
side
AB
c

θ 
cos
decreased to AB", BC" and AC".
AB
side
c
b
sideAC
For the angle

cos
sideAC
b
For the angle
AC is the hypotenuse
a
AC is the hypotenuse
a
a
b
sin
AB is the adjacent side
θ
b

a
 b
x a
AB is the adjacent side
 θ
sin
b
BC is the opposite side.
cos
c x
c
θ
c 
b
c
cos
c
b
θ
BC is the opposite side.
b
The ratios
b
The ratios
BC
side

tan
θ

BC
side

AB
side
tan


side
1
1
sin

or
θ.cosec
cosec
1
θ 
or
θ 
θ 

sin
1
1
θ
θ

cosec
sin = θ  = BC a BC AB a c tanθ cos θ θ θ sin  Exercise 1.6.26 1

sin

θ.cosec
or

θ 
or
θ 
cosec

cosec θ sin θ
1 1
cos θ  1 or sec θ  or cos . θ sec θ 1 
1
sec
θ
WORKSHOP CALCULATION & SCIENCE - CITS cos θ  sec θ or sec θ cos  θ θ or cos . θ sec θ 1 
cos
1 1
tan θ  1 or cot θ  1 or cot . θ tan θ 1 
θ

θ

tan
cot 
. θ
θ
θ

or
cot

tan
or

cot
1 

tan 
θ
By pythogoras theorem we have, AC = AB + BC 2
θ
θ

cot

tan
2
2
2
2
The six ratios between the sides have precise definitions. By pythogoras theorem we have, AC = AB + BC 2
The six ratios between the sides have precise definitions.
The six ratios between the sides have precise definitions.
BC Opposite side

Sine θ   BC Opposite side Sin θ
Sine θ AC  Hypotenuse  Sin θ

AC Hypotenuse
AB Adjacent side
Cosine θ   AB Adjacent side Cos θ


Cosine θ AC  Hypotenuse  Cos θ
AC Hypotenuse
BC Opposite side
Tangent θ   BC Opposite side Tan θ Dividing both sides of the equation by AC , we have

2
Tangent θ AB  Adjacent side  Tan θ Dividing both sides of the equation by AC , we have

2
AB Adjacent side 2 2 2
AC Hypotenuse AC = 2 AB + 2 BC 2

Cosecant θ  AC Hypotenuse  Cosec θ AC 2 AC AC = 2 AB AC + 2 BC
Cosecant θ BC  Opposite side  Cosec θ AC 2 AC 2 AC 2

BC Opposite side 2 2
AC Hypotenuse ⎡ AB⎤ ⎡ 2 BC⎤ 2
Secant θ  AC Hypotenuse  Sec θ = ⎢ ⎡ AB + ⎢ ⎡ ⎥ ⎤

⎥ ⎤
BC
AC⎦
AC⎦

Secant θ AB  Adjacent side  Sec θ ⎣ = ⎣ +
AB Adjacent side ⎢ ⎣ AC⎦ ⎥ ⎢ ⎣ AC⎦ ⎥
2
AB Adjacent side 1 = (cos θ) + (sin θ) 2
Cotangent θ  AB Adjacent side Cot θ 1 = (cos θ) + (sin θ) 2


2
θ

2
Cotangent BC  Opposite side  Cot θ sin θ + cos θ = 1
2
BC Opposite side sin θ + cos θ = 1
2
2
Relationship between the ratios
Relationship between the ratios Sine, Cosine, Tangent, Cosec, Sec and
Relationship between the ratios Cotangent are the six trigonometrical ratios
Sine, Cosine, Tangent, Cosec, Sec and
1
1
AC
Workshop Calculation & Science - Electronics Mechanic Exercise 1.6.26 Cotangent are the six trigonometrical ratios
θ 
Cosec

1
1
AC

Cosec θ BC  BC sin θ  Sin  2 2
and sin θθ θθ θ + cos θ θ θ θ θ = 1
θ = =
θ θ =
Trigonometry - Trigonometrical ratios BC AC BC sin θ tan θ = θ = θ θ = Sin  and sin θθ θθ θ + cos θ θ θ θ θ = 1
Cos 
θ = =
2
tan θ = θ =
2
AC Cos 
Dependency AC 1 1 61
sec θ    61
The sides of a triangle bear constant ratios for a given AB AB cos θ
definite value of the angle. That is, increase or decrease in AC
the length of the sides will not affect the ratio between them AB 1 1
unless the angle is changed. These ratios are trigonometrical cot θ   
ratios. For the given values of the angle a value of the ratios BC BC tanθ
AB
BC , AC , BC , AB , AB and AC
AB AB AC BC AC BC do not change even when sin θ  sideBC  a
the sides AB, BC, AC are increased to AB', BC' and AC' or sideAC b
decreased to AB", BC" and AC". cos θ  side AB  c
For the angle sideAC b
AC is the hypotenuse a
AB is the adjacent side sin θ  b  a x b  a
BC is the opposite side. cos θ c b c c
The ratios b
side BC
=  tan θ
side AB
1 1
sin θ  or cosec θ  or sin θ.cosec θ  1
cosec θ sin θ
1 1
cos θ  or sec θ  or cos . θ sec θ 1 
sec θ cos θ
1 1
tan θ  or cot θ  or cot . θ tan θ 1 
cot θ tan θ
By pythogoras theorem we have, AC = AB + BC 2
2
2
The six ratios between the sides have precise definitions.
BC Opposite side 82
Sine θ    Sin θ
AC Hypotenuse CITS : WCS - Electrical - Exercise 7
AB Adjacent side
Cosine θ    Cos θ
AC Hypotenuse
BC Opposite side
Tangent θ    Tan θ Dividing both sides of the equation by AC , we have
2
AB Adjacent side
AC 2 AB 2 BC 2
AC Hypotenuse = +
Cosecant θ    Cosec θ AC 2 AC 2 AC 2
BC Opposite side
2 2
AC Hypotenuse ⎡ AB⎤ ⎡ BC⎤
Secant θ    Sec θ = +
⎥
⎥
AB Adjacent side ⎢ AC⎦ ⎢ ⎣ AC⎦
⎣
2
AB Adjacent side 1 = (cos θ) + (sin θ) 2
Cotangent θ    Cot θ
BC Opposite side sin θ + cos θ = 1
2
2
Relationship between the ratios Sine, Cosine, Tangent, Cosec, Sec and
AC 1 1 Cotangent are the six trigonometrical ratios
Cosec θ   
BC BC sin θ Sin  2 2
θ = =
θ θ =
AC tan θ = θ = Cos  and sin θθ θθ θ + cos θ θ θ θ θ = 1
61