TRIGONOMETRIC FORMULAS APPLICABLE IN 9TH, 10TH STANDARD

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TRIGONOMETRIC FORMULAS APPLICABLE IN 9TH, 10TH STANDARD
https://dinesh51.blogspot.com
TRIGONOMETRY

sinθ=PerpendicularHypotenuseCosθ=BaseHypotenuseTanθ=PerpendicularBase
Cotθ=BasePerpendicularSecθ=HypotenuseBaseCosecθ=HypotenusePerpendicular
Sinθ=1CosecθCosθ=1SecθTanθ=1Cotθ
Tanθ=SinθCosθCotθ=CosθSinθ
TRIGONOMETRY
VALUE OF THE TRIGONOMETRIC FUNCTIONS WITH STANDARD ANGLE
Table 1
0o
30o
45o
60o
90o
180o
270o
360o
Sin
0
1/2
1/√2
√3/2
1
0
-1
0
Cos
1
√3/2
1/√2
1/2
0
-1
0
1
Tan
0
1/√3
1
√3
∞
0
∞
0
Cot
∞
√3
1
1/√3
0
∞
0
∞
Sec
1
2/√3
√2
2
∞
-1
∞
1
cosec
∞
2
√2
2/√3
1
∞
-1
∞

TRIGONOMETRIC IDENTITIES
Sin2θ+Cos2θ=1,Sin2θ=1−Cos2θ,Cos2θ=1−Sin2θ
Sec2θ−Tan2θ=1,Sec2θ=1+Tan2θ,Tan2θ=Sec2θ−1
Cosec2θ−Cot2θ=1,Cosec2θ=1+Cot2θ,Cot2θ=Cosec2θ−1
TRANSFORMATION OF ANGLES
 sin(90 - θ)   = cosθ,                        cos(90 - θ) =  sinθ,   
 tan(90 - θ)  = cotθ,                          cot(90 - θ) = tanθ,    
sec(90 - θ)   = cosecθ,                  cosec(90 - θ) = secθ
TRIGONOMETRIC FORMULAS APPLICABLE  10TH  ONWARD
MOVEMENT OF ANGLE IN DIFFERENT  QUADRANT
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TRIGONOMETRY

SIGN OF TRIGONOMETRIC RATIOS IN DIFFERENT QUADRANT
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COMPLETE  TRANSFORMATION OF ANGLES
 sin(90 - θ)   = cosθ,                        cos(90 - θ) =  sinθ,   
 tan(90 - θ)  = cotθ,                          cot(90 - θ) = tanθ,    
sec(90 - θ)   = cosecθ,                  cosec(90 - θ) = secθ

sin(90 + θ)   = cosθ,                        cos(90 + θ) =  -sinθ, 
 tan(90 + θ)  = -cotθ,                          cot(90 + θ) = -tanθ,    
sec(90 + θ)   = -cosecθ,                  cosec(90 + θ) = secθ

sin(180 - θ)   = sinθ,                          cos(180 - θ) =  -cosθ, 
 tan(180 - θ)  = -tanθ,                          cot(180 - θ) = -cotθ,    
sec(180 - θ)   = -secθ,                      cosec(180 - θ) = cosecθ

sin(180 + θ)   = -sinθ,                          cos(180 + θ) =  -cosθ, 
 tan(180 + θ)  = tanθ,                          cot(180 + θ) = cotθ,    
sec(180 + θ)   = -secθ,                      cosec(180 + θ) = -cosecθ

sin(270 - θ)   = -cosθ,                        cos(270 - θ) =  sinθ, 
 tan(270 - θ)  = cotθ,                          cot(270 - θ) = tanθ,    
sec(270 - θ)   = cosecθ,                  cosec(270 - θ) = secθ
DIFFERENT SYSTEMS OF ANGLES
CENTESIMAL SYSTEM
1Rightangle=100gradesor90o=100g
1grade=100minutesor1g=100′
1minute=100secondsor1′=100″
SEXAGESIMAL SYSTEM
1Rightangle=90o
1degree=60minuteor1o=60′
1minute=60secondor1′=60″
CIRCULAR SYSTEM
Radian Measure:- Angle made by an arc of unit length in a circle of unit radius is called one radian
OneRadian=UnitlengthofarcUnitRadius
θRadian=lengthofarcRadiusofCircleorθ=lrorl=rθ
RELATION BETWEEN DEGREE AND RADIAN
Angleatthecentreofthecircle=2πor360o
2πRadian=360o⇒πRadian=180o
1Radian=180π180×722=57o16′
180o=πRadian⇒1o=πRadian=227×180Radian=0.01746Radian
tan(A+B)=tanA+tanB1−tanAtanB
tan(A−B)=tanA−tanB1+tanAtanB
cot(A+B)=cotAcotB−1cotA+cotB
cot(A−B)=cotAcotB+1cotA−cotB
TRIGONOMETRY-shikhakaushal.blogspot.com
TRIGONOMETRIC FORMULAS WITH MULTIPLE ANGLE
Sin2θ=2SinθCosθOR2tanθ1+tan2θ
Cos2θ=Cos2θ−Sin2θ,or2Cos2θ−1,or1−2Sin2θor1−tan2θ1+tan2θ
tan2θ=2tanθ1−tan2θ
Sin3θ=3Sinθ−4Sin3θ
Cos3θ=−(3Cosθ−4Cos3θ)or(4Cos3θ−3Cosθ)
Tan3θ=3tanθ−tan3θ1−3tan2θ
Sin2θ=1−cos2θ2,Sin24θ=1−cos8θ2
Cos2θ=1+cos2θ2,Cos24θ=1+cos8θ2
tan2θ=1−cos2θ1+cos2θ,tan24θ=1−cos8θ1+cos8θ
TRIGONOMETRIC FORMULAS WITH SUB-MULTIPLE ANGLE
Sinθ=2sinθ2cosθ2,
Sinθ=2sinθ2cosθ2,or2tanθ21+tan2θ2
Cos2θ=Cos2θ2−Sin2θ2,or2Cos2θ2−1,or1−2Sin2θ2or1−tan2θ21+tan2θ2
Tanθ=2tanθ21−tan2θ2
Sin2θ2=1−cosθ2,Cos2θ2=1+cosθ2
tan2θ2=1−cosθ1+cosθ,
A, B FORMULAS:-
2SinACosB=Sin(A+B)+Sin(A−B)
2CosASinB=Sin(A+B)−Sin(A−B)
2CosACosB=Cos(A+B)+Cos(A−B)
2SinASinB=−Cos(A+B)+Cos(A−B)
C,D FORMULAS:- 
SinC+SinD=2Sin(C+D2)Cos(C−D2)
SinC−SinD=2Cos(C+D2)Sin(C−D2)
CosC+CosD=2Cos(C+D2)Cos(C−D2)
CosC−CosD=−2Sin(C+D2)Sin(C−D2)
SOME SPECIAL FORMULAS
Tan(45+θ)=1+tanθ1−tanθ
Tan(45−θ)=1−tanθ1+tanθ
Sin2A−Sin2B=Sin(A+B)Sin(A−B)
Cos2A−Cos2B=Sin(A+B)Sin(A−B)
PERIODIC FUNCTIONS
All trigonometric ratios are periodic functions
Sin(x+2π)=Sin(x+4π)=Sin(x+6π)=Sin(x+8π)=..........
Cos(x+2π)=Cos(x+4π)=Cos(x+6π)=Cos(x+8π)=..........
Sec(x+2π)=Sec(x+4π)=Sec(x+6π)=Sec(x+8π)=..........
Cosec(x+2π)=Cosec(x+4π)=Cosec(x+6π)=Cosec(x+8π)=..........
Onchangingxto2nπ,sinx,cosx,secx,cosecxallremainunchanged.
sinx,cosx,secx,cosecxallareperiodicfunctionsandtheirperiodis2π
tanx=tan(x±π)=tan(x±2π)=tan(x±3π)=........
cotx=cot(x±π)=cot(x±2π)=cot(x±3π)=........
tanxandcotxareperiodicandtheirperiodisπ
PRINCIPAL SOLUTION OF TRIGONOMETRIC FUNCTIONS
Solutionsoftrigonometricequationsforwhich0≤x≤2πarecalledprincipalsolutions
PERIODIC VALUES:-
It is the least value which when added to the given function so that the value of that function remain unchanged.
TRIGONOMETRIC EQUATIONS:-
Equations involving trigonometric functions of a variable are called trigonometric equations.
GENERAL SOLUTIONS OF TRIGONOMETRIC EQUATIONS:-
Ifsinx=0,thenx=nπ
Ifcosx=0,thenx=(2n+1)π2
Iftanx=0,thenx=nπ
Ifsinx=sinythenx=nπ+(−1)ny
Ifcosx=cosy,thenx=2nπ±y
Iftanx=tany,thenx=nπ+ywww.shikhakaushal.blogspot.com
Sin(A+B)=SinACosB + CosASinB
Sin(A-B)=SinACosB - CosASinB
Cos(A+B)=CosACosB - SinA SinB
Cos(A-B)=CosACosB + SinA SinB

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