TRIGONOMETRY

1800 = πC = 200g ( 1c ≈57017’45” , 10 = 0.01745c )

sin2θ +cos2θ = 1
sec2θ – tan2θ = 1
cosec2θ – cot2θ = 1
sin4θ + cos4θ = 1 - 2 sin2θ cos2θ
sin6θ + cos6θ = 1 – 3 sin2θ cos2θ
tan2θ – sin2θ = tan2θ sin2θ
cot2θ – cos2θ = cot2θ cos2θ
sec2θ + cosec2θ = sec2θ cosec2θ

sin (A + B) = sin A cos B + cos A sin B

sin (A – B) = sin A cos B – cos A sin B
cos (A + B) = cos A cos B – sin A sin B
cos (A – B) = cos A cos B + sin A sin B
sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A
cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A
tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
tan (A – B) = (tan A – tan B) / (1 + tan A tan B)
cot (A + B) = (cot A cot B – 1)/(cot A + cot B)
cot (A – B) = (cot A cot B + 1)/(cot B – cot A)
sin C + sin D = 2 sin (C + D)/2 cos (C – D)/2
sin C – sin D = 2 cos (C + D)/2 sin (C – D)/2
cos C + cos D = 2 cos (C + D)/2 cos (C – D)/2
cos C – cos D = – 2 sin (C + D)/2 sin (C – D/2
2 sin A cos B = sin (A + B) + sin (A – B)
2 cos A sin B = sin (A + B) – sin (A – B)
2 cos A cos B = cos(A + B) + cos(A – B)
2 sin A sin B = cos (A – B) – cos (A + B)
sin 2A = 2 sin A cos A
cos 2A = cos2 A – sin2 A = 1 – 2 sin2A = 2 cos2 A – 1
cos2A = (1 + cos 2A)/2
sin2 A = (1 – cos 2A)/2
tan 2A = 2 tan A/(1 – tan2 A)
sin 2A = 2 tan A/(1 + tan2 A)
cos 2A = (1 – tan2 A)/(1 + tan2 A)
sin (A + B + C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C – sin A sin B sin C
cos (A + B + C) = cos A cos B cos C – sin A sin B cos C – sin A cos B sin C – cos A sin B sin C
sin 3A = 3 sin A – 4 sin3 A
cos 3A = 4 cos3A – 3 cos A
tan (θ 1 + θ 2 + θ 3 + ______) = (S1 – S3 + S5 – S7 + _____)/(1 – S2 + S4 – S6 + _____)
where S1 = ∑ tan θ 1 , S2 = ∑ tan θ 1 tan θ 2 , S3 = ∑ tan θ 1 tan θ 2 tan θ 3

tan 3A = (3 tan A – tan3 A)/(1 – 3 tan2 A)

 sin 3 A
sinA sin(600+A) sin(600-A) =
4
cos 3A
cosA cos(600+A) cos(600-A) =
4
tanA tan(600+A) tan(600-A) = tan3A
cotA cot(600+A) cot(600-A) = cot3A
If A + B + C = π , then
tan A + tan B + tan C = tan A tan B tan C
cot A cot B + cot B cot C + cot C cot A = 1

tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2 = 1
cot A/2 + cot B/2 + cot C/2 = cot A/2 cot B/2 cot C/2
If A + B + C = π /2, then
cot A + cot B + cot C = cot A cot B cot C
tan A tan B + tan B tan C + tan C tan A = 1

sin 15° = (√ 3 – 1)/2√ 2 = cos 750

cos 15° = (√ 3 + 1)/2√ 2 = sin 750
sin 18° = (√ 5 – 1)/4 = cos 720
cos 18° = √ 10+2 √ 5 /4 = sin 720
sin 36° = √ 10−2 √ 5 /4 =cos 540
cos 36° = (√ 5 + 1)/4 = sin 540

sin 22.50 =
√ √2−1
2 √2
= cos 67.50

cos 22.50 =
√ √2+1
tan 22.50 = √ 2 - 1
2 √2
= sin 67.50

= cot 67.50
cot 22.50 = √ 2 + 1 = tan 67.50

sin α + sin (α + β ) + sin (α + 2 β ) + ______ n terms = sin (n β /2) /sin ( β /2) sin (α + (n – 1) β /2)
cos α + cos (α + β ) + cos (α + 2 β ) + _____ n terms = sin (n β /2)/sin ( β /2) cos(α + (n – 1) β /2)
– √ a2 +b 2 +c < a sin θ + b cos θ +c < √ a2 +b 2 + c
3
sin2θ + sin2(X+θ) + sin2(X-θ) = where X = 600 or 1200 or 2400
2
3
cos2θ + cos2(X+θ) + cos2(X-θ) = where X = 600 or 1200 or 2400
2

π π 1+tanθ cosθ +sinθ

tan( + θ ) = cot ( - θ) = =
4 4 1−tanθ cosθ−sinθ

π π 1−tanθ cosθ−sinθ
tan( - θ ) = cot ( + θ) = =
4 4 1+tanθ cosθ +sinθ

π π π π
tan( - θ ) tan ( + θ) = cot( - θ ) cot ( + θ) = 1
4 4 4 4
 π π
tan( + θ ) + tan ( - θ) = 2sec2θ
4 4

π π
tan( + θ ) - tan ( - θ) = 2tan2θ
4 4
cos θ +cos(1200+θ) + cos(1200 - θ) = 0 ::: cos(1200+θ ) = cos(2400-θ ); cos(1200- θ) = cos(2400+θ )

sinA=sinB then A=B or A+B=1800

cosA=cosB then A=B or A+B=0 or A+B=3600

tanA=tanB then A=B or |A-B|=1800

If A+B = 900 then tanA – tanB = 2 tan( A – B)

tanθ+tan(600+ θ)+ tan(1200+ θ) = 3tan3θ

cotθ+cot(600+ θ)+ cot(1200+ θ) = 3 cot3θ

tanθ tan(600+θ) + tanθ tan(1200+θ)+ tan(600+ θ) tan(1200+θ) = - 3

cotθ cot (600+θ) + cotθ cot(1200+θ)+ cot(600+ θ) cot(1200+θ) = - 3

If A+B=900 then sin2A + sin2B = 1 and cos2A + cos2 B = 1

sin ⁡( A+ B)
tanA + tanB =
cosAcosB

sin ⁡( A−B)
tanA - tanB =
cosAcosB
tanA + cotA = 2cosec 2A
tanA – cotA = - 2cot2A
A A
√(1+ sinA ) = ± ( sin 2 + cos 2 )
A A
√(1−sinA ) = ± ( sin 2 −cos 2 )
2π
Period of a sin(± kθ +b ) +c is
|k|

2π
Period of a cos(± kθ +b ) +c is
|k|
 2π
Period of a cosec(± kθ +b ) +c is
|k|

2π
Period of a sec(± kθ +b ) +c is
|k|

π
Period of a cot(± kθ +b ) +c is
|k|

π
Period of a tan(± kθ +b ) +c is
|k|

Period of nx - [ nx ] is 1/|n|

sinθ = k , -1≤ k ≤ 1 then θ = nπ + (-1)n α where αЄ [ −π π

,
2 2 ]
cosθ = k , -1≤ k ≤ 1 then θ = 2nπ ± α where αЄ [ 0 , π ]

tanθ = k , k Є R then θ = nπ + α where αЄ ( −π2 , π2 )

For simultaneous equations θ = 2nπ + α where α Є [ 0 , 2π )

sin2θ = k or cos2θ = k or tan2θ = k then θ = nπ ±α where α Є 0 , [ ]

π
2

INVERSE TRIGONOMETRIC FUNCTIONS

Function : Domain Range

π π
Sin-1x or arc Sine : [ -1 , 1] [- , ]
2 2

Cos-1x or arc Cos : [ -1 , 1] [ 0, π ]

π π
Tan-1x or arc Tan : R (- , )
2 2

Cot-1x or arc Cot : R (0 , π)

π
Sec-1x or arc Sec : (- ∞,-1]∩[1,∞) [ 0 , π ] –{ }
2
π π
Cosec-1x or arc Cosec : (- ∞,-1]∩[1,∞) [- , ] –{ 0 }
2 2

Sin-1(-x) = - Sin-1(x) Tan-1(-x) = -Tan-1(x) Cosec-1(-x) = -Cosec-1(x)

Cos-1(-x)=π-Cos-1(x) Sec-1(-x)=π-Sec-1(x) Cot-1(-x)=π-Cot-1(x)

π π π
Sin-1(x)+Cos-1(x) = Tan-1(x)+Cot-1(x) = Sec-1(x)+Cosec-1(x) =
2 2 2

1 1 1
Cosec-1(x)=Sin-1( ) Sec-1(x)=Cos-1( ) Cot-1(x) = Tan-1( ) x›0
x x x

1
= Π + Tan-1( ) x‹0
x

π
Sin-1x+Cos-1x = x Ɛ [ −1 ,1 ]
2

π
Tan-1x+Cot-1x = xƐR
2

π
Sec-1x+Cosec-1x = x Ɛ ¿−∞ ,−1 ¿ ¿ υ ¿
2

Tan-1x+Tan-1y = Tan-1 ( 1−xy

x+ y
) ; x›0, y›0 , xy‹1
= Tan-1 ( 1−xy
x+ y
) +π ; x›0, y›0 , xy›1