TRIGONOMETRIC FORMULAS APPLICABLE IN 9TH, 10TH STANDARD
TRIGONOMETRIC FORMULAS APPLICABLE IN 9TH, 10TH STANDARD
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
TRIGONOMETRY
SIGN OF TRIGONOMETRIC RATIOS IN DIFFERENT QUADRANT
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
Sin2θ=2SinθCosθOR2tanθ1+tan2θ
TRIGONOMETRY-shikhakaushal.blogspot.com
TRIGONOMETRIC FORMULAS WITH MULTIPLE ANGLE
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.
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
TRIGONOMETRIC FORMULAS WITH COMPOUND ANGLE
Sin(A+B)=SinACosB + CosASinB
Sin(A-B)=SinACosB - CosASinB
Cos(A+B)=CosACosB - SinA SinB
Cos(A-B)=CosACosB + SinA SinB
