TRIGNOMENTARY
Trignomentary is area of math
where relationship among angle are studied in triangle.
Relationship among Base, perpendicular and
hypotenuse
Sin
A=P/H , cosec A=H/P since ( sin A=1/cosec A ,
cosec A=1/sin A)
Cos
A=B/H , sec A=H/B since ( cos A=1/sec A , sec
A=1/ cos A)
Tan
A=P/B , cot A=B/P since (tan A=1/cot A , cot A=1/tan A)
Quadrant System :
In
above table…
·
(90+ A),(90-A)
and (270-A),(270+A) Trigonometric function will change, and given
below.
·
(180 +A),(180-A) and (270+A),(270- A) ) Trigonometric function will remain the same,
and is given below.
·
1 st quadrant
table ranges [0-90 degree] in which all Trigonometric function have positive result and their
conversion table given below.
.i.e
Sin(90-A)=cos A
cos(90-A)=sin A
tan(90-A)=cot A
cot(90-A)=tan A
sec(90-A)=cosec A
cosec(90-A)= sec A
·
2 nd quadrant
table ranges [90-180 degree] ,in which
only [sin ,cosec ]Trigonometric function
have positive result and their conversion table given below.
.i.e
Sin(90 +A)= + cos
A
Sin(180-A)= + Sin A
cos(90+A)= - sin
A
cos(180-A)= -cos A
tan(90+A)= - cot
A
tan (180-A)= - tan A
cot(90+A)= - tan
A
cot(180-A)= -cot A
sec(90+A)= - cosec
A
sec(180-A)= -sec A
cosec(90+A)= sec A cosec (180-A)= cosec A
·
3 rd quadrant
table ranges[180- 270 degree], in which
only [tan,cot ]Trigonometric function have
positive result and their conversion table given below.
.i.e
Sin(270-A)= - cos Sin(180+A)=Sin A
cos(270-A)= - sin
A cos(180+A)= -cos A
tan(270-A)= cot
A tan (180+A)= tan A
cot(270-A)= tan
A cot(180+A)= cot A
sec(270-A)= - cosec
A sec(180+A)= -sec A
cosec(270-A)= -sec
A cosec (180+A)= -cosec A
·
4 th quadrant
table ranges[270-360 degree], in
which only [cos,sec, ]Trigonometric function have positive result and their
conversion table given below.
.i.e
Sin(270 +A)= - cos
A Sin(360-A)= - Sin A
cos(270 +A)= sin
A cos(360-A)= cos A
tan(270 +A)= - cot
A tan (360-A)= - tan A
cot(270 +A)= -tan
A cot(360-A)= - cot A
sec(270 +A)= cosec
A sec(360-A)= sec A
cosec(270 +A)= -sec
A cosec (360-A)= -cosec A
Angle Value Table :
Trick to remember::
For sin A remember any angle value (30,45,60,90
degree), in this case I have chosen 30
degree. Now let us assume the triangle is Right triangle, then
Sin
A=perpendicular/ base (since
remember only one value let suppose angle A=30 )
Sin 30= ½
Thus
Perpendicular
=1 , hypotenuse =2
Since triangle is right triangle thus
(Perpendicular)2
+ (base)2 =(hypotenuse)2
12 + (base)2 =
22
Base=
Sqrt(3) [ here sqrt= square root]
Thus we can conclude
value of other value of “Sin A”.
Similarly we can conclude other
value of (cos A, tan A, etc) .
Elementary Trigonometric
Identities:
1.
(sin A) 2 + (cos A)2 = 1
OR sin2 A + cos2 A=1
2.
(sec A)2 = 1 +
(tan A)2 OR sec2 A=1+
tan2 A
3.
(cosec A)2 = 1 +(cot A)2 OR cosec2 A=1+ cot 2A
Some Trigonometric
Identities:
1.
Sin(A+B) = sin A * cos B
+ cos A * sin B
2.
Sin(A-B) = sin A * cos B
- cos A * sin B
3.
Cos(A+B) = cos A * cos
B - sin A* sin B
4.
Cos(A-B) = cos A * cos
B + sin A* sin B
5.
Tan(A +B) = [tan A +
tan B] / [1-tan A * tan B]
6.
Tan(A -B) = [tan A -
tan B] / [1 + tan A * tan B]
7.
Cot(A+B) = [cot A * cot
B – 1 ]/[cot A + cot B]
8.
Cot(A-B) = [cot A * cot
B + 1 ]/[cot A - cot B]
Some other:
1.
2 sin A sin B = cos(A-B) – cos( A+B)
2.
2 cos A cos B= cos(A+B) + cos( A-B)
3.
2 sin A cos B=sin(A+B) + sin(A-B)
4.
2 cos A Sin B = sin(A+B) - sin(A-B)
5.
Sin C + sin D= 2 sin {(C+D)} cos(C-D)
2
2
6.
Sin C - sin D= 2 cos {(C+D)} sin(C-D)
2 2
7.
Cos C + cos
D= 2 cos {(C+D)} cos(C-D)
2 2
8.
Cos C
– cos D= 2 cos {(C+D)} sin(D-C)
2 2
Some other:
1.
Sin
2A = 2 sin A cos B
=2 tan
A
1 + tan2 A
2.
Cos 2A=
cos2 A - sin2
A = 2 cos2 A -1 = 1- 2 sin2
A
= 1 - tan2 A
1 +
tan2 A
3.
Tan 2A=
2 tan A
1-
Tan2 A
4.
Sin 3A= 3 sin A – 4 sin3
A
5.
Cos 3A=4 cos3
A - 3 cos A
6.
Tan 3A= 3 tan A – tan3 A
1-
3 tan2 A
Relationship between Radian and degree :
Π radian
= 180°
Relationship between length of arc (l), radius(r), central angle
(A)
If there are 2
circle and central angle are formed at first and second circle are A1 ,A2 and , radius are r1 r1
, and length of arc l1 l2.
Now
A1= l1/ r1
and A2= l2/ r2
Trick to find Relationship angle between hand of clock :
11/2
* minute =30 *hour + angle
Min and Max range Trigonometric function:
1.
Sin A [-1,1 ]
2.
Cos A [ -1,1
]
3.
Tan A [ - ∞, + ∞ ]
4.
Cosec A [ - ∞,-1 ] U [1, ∞]
5.
Sec A [
- ∞,-1 ] U
[1, ∞]
6.
Cot A [- ∞, + ∞ ]
Min and Max range Trigonometric Equation :
1.
a sinɸ ± b cosɸ,
a sinɸ
± b sinɸ,
a cosɸ
± b cosɸ
Maximum value = √ (a2 + b2)
Minimum value = – √ (a2 + b2)
Maximum value = √ (a2 + b2)
Minimum value = – √ (a2 + b2)
2.
a sin2ɸ + b cosec2ɸ,
a cos2ɸ + b sec2ɸ,
a tan2ɸ + b cot2ɸ
Minimum value = 2√(ab)
The maximum value =∞
Minimum value = 2√(ab)
The maximum value =∞
3.
a sin2ɸ + b cos2ɸ
If a > b, Maximum value = a and Minimum value = b
If a < b, Maximum value = b and Minimum value = a
If a > b, Maximum value = a and Minimum value = b
If a < b, Maximum value = b and Minimum value = a
4.
a sinɸ + b cosecɸ , a cosɸ + b secɸ , a tanɸ + b cotɸ
Minimum
value = 2√(ab)
The maximum value =∞
The maximum value =∞
5.
a secɸ + b cosecɸ
Maximum value =[ √ (a +√ (b ]2
Minimum value = ∞
Minimum value = ∞
Shortcut for solution of
questions :
·
If the question
have equal sign (=) .i.e. “LHS of equal
side “ as “RHS of equal side “ . Then in that case put
ɸ value such that both hand
side are equal, if so then put ɸ value in given equation.
·
If the question
have only equation then put ɸ value in equation , also
put ɸ value in the given
equation to find the value and verify the vale from give objective typo.
Height and distance
In height and distance we will talk more about Right triangle .i.e.
more question asked from Right triangle.
Angle of elevation :
The angle
of elevation of an object as
seen by an observer is the angle between the horizontal and the line from the
object to the observer's eye (the line of sight).
Angle of depression :
If the object is below the level of the observer, then the angle
between the horizontal and the observer's line of sight is called the angle of depression.
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RELATION BETWEEN TWO:
Some short trick:
·
Ease
way to solve the question is angle of elevation. Therefore if angle of elevation
is given then ok ,go for that.
·
If
angle of depression is given assume it as angle of elevation and make solution for that.
·
In
right angle triangle , suppose 60°
,30° ,
·
In
right angle triangle , suppose 45°
,45° ,
·
When D1 and D2 height of towers, and x is distance between
then
.x=√ (D1
.D2)
·
if
ɸ1 and ɸ2 are two angle of elevation from
same level and “ d ” is distance between them and “h “ is height then
.d=h(cot ɸ1 – cot ɸ2 )
·
if ɸ1 and ɸ2 are two angle of elevation from
different level and “ h ” is height
between them and “H “ is height from base to object that has to be
seen then
H= h cot ɸ1
cot ɸ1 – cot ɸ2
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