TRIGONOMETRIC RATIO & IDENTITIES
sin (π – θ) = sin θ
1. Relation between systems of measurement
of angles :
D
G
2C
=
=
90 100
π
2. Fundamental Trigonometric Identities :
(i)
cos (π – θ) = – cos θ
tan (π – θ) = – tan θ
cot (π – θ) = – cot θ
sec (π – θ) = – sec θ
cosec (π – θ) = cosec θ
sin (π + θ) = – sin θ
cos (π + θ) = – cos θ
tan (π + θ) = tan θ
sin 2 θ + cos 2 θ = 1
cot (π + θ) = cot θ
sec (π + θ) = – sec θ cosec (π + θ) = – cosec θ
(ii) 1 + tan 2 θ = sec 2 θ
FG 3π − θIJ = − cos θ cos FG 3π − θIJ = − sin θ
H2 K
H2 K
F 3π − θIJ = cot θ cot FG 3π − θIJ = tan θ
tan G
H2 K
H2 K
F 3π I
F 3π I
sec G − θJ = −cosec θ cosec G − θJ = − sec θ
H2 K
H2 K
F 3π + θIJ = − cos θ cos FG 3π + θIJ = sin θ
sin G
H2 K
H2 K
F 3π + θIJ = − cot θ cot FG 3π + θIJ = − tan θ
tan G
H2 K
H2 K
F 3π + θIJ = cosec θ
sec G cot
tan
B1 A − sin B
±Btan
cos ( A + B)H cos
( AAKA−cot
)B
= ∓cos
cot
tan ( A ± B) =2
tan AB
B ±IAcot
3 π tan
F1∓cot
cosec G
+ θJ = − sec θ
H2 K
sin
(iii) 1 + cot 2 θ = cos ec 2 θ
3. Sign of trigonometrical ratios or functions :
2
ADD TO MEMORY
“Add Sugar To Coffee”
cos (2π – θ) = cos θ
tan (2π – θ) = – tan θ
cot (2π – θ) = – cot θ
cosec (2π – θ) = – cosec θ
cos (– θ) = cos θ
cot (– θ) = – cot θ
sec (– θ) = sec θ
FG π − θIJ = cos θ
Fπ I
cos G − θJ = sin θ
H2 K
H2 K
Fπ I
Fπ I
tan G − θJ = cot θ
cot G − θJ = tan θ
H2 K
H2 K
Fπ I
Fπ I
sec G − θJ = cosec θ cosec G − θJ = sec θ
H2 K
H2 K
Fπ I
Fπ I
sin G + θJ = cos θ
cos G + θJ = − sin θ
H2 K
H2 K
Fπ I
Fπ I
tan G + θJ = − cot θ
cot G + θJ = − tan θ
H2 K
H2 K
Fπ I
Fπ I
sec G + θJ = −cosec θ cosec G + θJ = sec θ
H2 K
H2 K
sin (2π – θ) = – sin θ
sec (2π – θ) = sec θ
4. Trigonometric ratios of allied angles :
sin (– θ) = – sin θ
tan (– θ) = – tan θ
cosec (– θ) = – cosec θ
2
sin (2π + θ) = sin θ
cos (2π + θ) = cos θ
tan (2π + θ) = tan θ
cot (2π + θ) = cot θ
sec (2π + θ) = sec θ
sin
cosec (2π + θ) = cosec θ
5. Sum and Difference formulae :
(i)
sin ( A ± B) = sin A cos B ± cos A sin B
(ii) cos ( A ± B) = cos A cos B ∓ sin A sin B
(iii)
(iv)
(v) sin ( A + B) sin ( A − B) = sin2 A − sin2 B