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Trigonometry formula
Angle
00
300
450
600
900
sin
0
1
2
1
1
√2
1
√3
2
1
2
√3
∞
1
1
0
∞
1
cos
1
tan
0
√3
2
1
cot
∞
√3
√3
sec
cosec
√2
1
1
2
√2
√3
2
∞
√3
2
√2
2
0
√3
sin (900 − 𝜃) = cos 𝜃
cos (900 − 𝜃 ) = sin 𝜃
tan (900 − 𝜃) = cot 𝜃
cot (900 − 𝜃 ) = tan 𝜃
sec (900 − 𝜃 ) = cosec 𝜃
cosec (900 − 𝜃) = sec 𝜃
sin (900 + 𝜃) = cos 𝜃
cos (900 + 𝜃 ) = − sin 𝜃
tan (900 + 𝜃) = − cot 𝜃
cot (900 + 𝜃 ) = − tan 𝜃
sec (900 + 𝜃 ) = − cosec 𝜃
cosec (900 + 𝜃) = sec 𝜃
sin (1800 − 𝜃) = sin 𝜃
cos (1800 − 𝜃 ) = − cos 𝜃
tan (1800 − 𝜃) = − tan 𝜃
cot (1800 − 𝜃 ) = − cot 𝜃
sec (1800 − 𝜃 ) = − sec 𝜃
cosec (1800 − 𝜃) = cosec 𝜃
sin (1800 + 𝜃) = − sin 𝜃
cos (1800 + 𝜃 ) = − cos 𝜃
tan (1800 + 𝜃) = tan 𝜃
cot (1800 + 𝜃 ) = cot 𝜃
sec (1800 + 𝜃 ) = − sec 𝜃
cosec (1800 + 𝜃) = − cosec 𝜃
sin (3600 − 𝜃) = sin(−𝜃) = − sin 𝜃
cos (3600 − 𝜃 ) = cos(−𝜃) = cos 𝜃
tan (3600 − 𝜃) = tan(−𝜃) = − tan 𝜃
cot (3600 − 𝜃 ) = cot(−𝜃) = − cot 𝜃
sec (3600 − 𝜃 ) = sec(−𝜃) = sec 𝜃
cosec (3600 − 𝜃) = cosec(−𝜃) =
−𝑐𝑜𝑠𝑒𝑐 𝜃
sin (3600 + 𝜃) = sin 𝜃
cos (3600 + 𝜃 ) = cos 𝜃
tan (3600 + 𝜃) = tan 𝜃
cot (3600 + 𝜃 ) = cot 𝜃
sec (3600 + 𝜃 ) = sec 𝜃
cosec (3600 + 𝜃) = cosec 𝜃
(1) 𝑠𝑖𝑛 2 𝐴 + 𝑐𝑜𝑠 2 𝐴 = 1, 𝑠𝑒𝑐 2 𝐴 = 1 + 𝑡𝑎𝑛 2 𝐴 , 𝑐𝑜𝑠𝑒𝑐 2 𝐴 = 1 + 𝑐𝑜𝑡 2 𝐴
(2) 𝑐𝑜𝑠𝑒𝑐 𝐴 =
1
sin 𝐴
; sec 𝐴 =
1
cos 𝐴
; tan 𝐴 =
1
cot 𝐴
; cot 𝐴 =
sin 𝐴
cos 𝐴
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(3) sin 2𝐴 = 2 sin 𝐴 cos 𝐴
(4) sin 2𝐴 =
2 tan 𝐴
1+𝑡𝑎𝑛2𝐴
𝐴
𝐴
2
2
(5) sin 𝐴 = 2 sin cos
𝐴
(6) sin 𝐴 =
2 tan 2
1+𝑡𝑎𝑛2
𝐴
2
(7) cos 2𝐴 = 𝑐𝑜𝑠 2 𝐴 − 𝑠𝑖𝑛 2 𝐴
(8) cos 2𝐴 = 1 − 2𝑠𝑖𝑛 2 𝐴
(9) cos 2𝐴 = 2𝑐𝑜𝑠 2 𝐴 − 1
(10) cos 2𝐴 =
1−𝑡𝑎𝑛2𝐴
1+𝑡𝑎𝑛2𝐴
𝐴
𝐴
2
2
(11) cos 𝐴 = 𝑐𝑜𝑠 2 − 𝑠𝑖𝑛 2
(12) cos 𝐴 = 1 − 2 𝑠𝑖𝑛 2
𝐴
2
𝐴
(13) cos 𝐴 = 2 𝑐𝑜𝑠 2 − 1
2
(14) cos 𝐴 =
𝐴
2
𝐴
1+𝑡𝑎𝑛2
2
1−𝑡𝑎𝑛2
2 𝑡𝑎𝑛𝐴
(15) tan 2𝐴 =
1−𝑡𝑎𝑛2𝐴
𝐴
(16) tan 𝐴 =
2 tan 2
𝐴
1−𝑡𝑎𝑛2 2
(17) cot 2𝐴 =
𝑐𝑜𝑡 2𝐴−1
2 cot 𝐴
𝐴
(18) cot 𝐴 =
𝑐𝑜𝑡 2 2 −1
𝐴
2 cot 2
(19) sin 3𝐴 = 3 sin 𝐴 − 4 𝑠𝑖𝑛 3 𝐴
𝐴
𝐴
3
3
(20) sin 𝐴 = 3 sin − 4 𝑠𝑖𝑛 3
(21) cos 3𝐴 = 4 𝑐𝑜𝑠 3 𝐴 − 3 cos 𝐴
𝐴
𝐴
3
3
(22) cos 𝐴 = 4 𝑐𝑜𝑠 3 − 3 cos
(23) tan 3𝐴 =
3 tan 𝐴−𝑡𝑎𝑛3𝐴
1−3 𝑡𝑎𝑛2𝐴
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(24) cot 3𝐴 =
𝑐𝑜𝑡 3𝐴−3 cot 𝐴
3 𝑐𝑜𝑡 2𝐴−1
(25) sin(𝐴 + 𝐵 ) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
(26) sin(𝐴 − 𝐵 ) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
(27) cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
(28) cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
tan 𝐴+tan 𝐵
(29) tan(𝐴 + 𝐵 ) =
1−tan 𝐴 tan 𝐵
(30) cot(𝐴 + 𝐵 ) =
cot 𝐴 cot 𝐵−1
(31) cot(𝐴 − 𝐵 ) =
cot 𝐴 cot 𝐵+1
cot 𝐵+cot 𝐴
cot 𝐵−cot 𝐴
(32) 𝑠𝑖𝑛 (𝐴 + 𝐵 ) sin(𝐴 − 𝐵 ) = 𝑠𝑖𝑛 2 𝐴 − 𝑠𝑖𝑛 2 𝐵
(33) 𝑐𝑜𝑠 (𝐴 + 𝐵 ) cos (𝐴 − 𝐵 ) = 𝑐𝑜𝑠 2 𝐴 − 𝑐𝑜𝑠 2 𝐵
(34) sin 𝐶 + sin 𝐷 = 2 sin
𝐶+𝐷
(35) sin 𝐶 − sin 𝐷 = 2 cos
2
𝐶+𝐷
2
(36) cos 𝐶 + cos 𝐷 = 2 cos
(37) cos 𝐶 − cos 𝐷 = 2 sin
cos
sin
𝐶+𝐷
2
𝐶+𝐷
2
𝐶−𝐷
2
𝐶−𝐷
2
cos
sin
𝐶−𝐷
2
𝐷−𝐶
2
(38) 2 sin 𝐴 cos 𝐵 = sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)
(39) 2 cos 𝐴 sin 𝐵 = sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵)
(40) 2 cos 𝐴 cos 𝐵 = cos(𝐴 + 𝐵) + cos (𝐴 − 𝐵)
(41) 2 sin 𝐴 sin 𝐵 = cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)
𝐴
𝐴
2
2
(42) sin 𝐴 = 2 sin cos
(43)
(44)
sin 𝐴
𝑎
𝑎
sin 𝐴
=
=
sin 𝐵
𝑏
𝑏
sin 𝐵
=
=
sin 𝐶
𝑐
𝑐
sin 𝐶
(45) 𝑎 = 𝑏 cos 𝐶 + 𝑐 cos 𝐵, 𝑏 = 𝑐 cos 𝐴 + 𝑎 cos 𝐶, 𝑐 = 𝑎 cos 𝐵 + 𝑏 cos 𝐴
(46) cos 𝐴 =
𝑏2 +𝑐 2−𝑎 2
𝑐 2+𝑎 2 −𝑏2
2 𝑏𝑐
2𝑐𝑎
, cos 𝐵 =
, cos 𝐶 =
𝑎 2+𝑏2−𝑐 2
2 𝑎𝑏
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(47) 𝑎2 = 𝑏 2 + 𝑐 2 − 2 𝑏𝑐 cos 𝐴
(48) 𝑏 2 = 𝑐 2 + 𝑎2 − 2 𝑐𝑎 cos 𝐴
(49) 𝑐 2 = 𝑎2 + 𝑏 2 − 2 𝑎𝑏 cos 𝐴
𝐴
(𝑠−𝑏)(𝑠−𝑐)
2
𝑏𝑐
𝐴
𝑠(𝑠−𝑎)
2
𝑏𝑐
(50) sin = √
(51) cos = √
𝐴
(52) tan
2
=√
(53) sin 2𝐴 =
(54) tan
(55) tan
(56) tan
𝐴−𝐵
2
𝐵−𝐶
2
𝐶−𝐴
2
𝑠(𝑠−𝑎)
𝑏𝑐
=
=
=
𝑎+𝑏
𝑏−𝑐
𝑏+𝑐
𝑐−𝑎
𝑐+𝑎
cot
(59) 𝑟 =
cot
cot
, 𝑅=
𝐴
2
𝑠(𝑠−𝑐)
2
𝑎𝑏
(𝑠−𝑐)(𝑠−𝑎)
𝑠(𝑠−𝑏)
, sin 2𝐶 =
, tan
𝐶
2
=√
(𝑠−𝑎)(𝑠−𝑏)
𝑠(𝑠−𝑐)
2∆
𝑎𝑏
𝐵
2
𝑏
2 sin 𝐵
= (𝑠 − 𝑎) tan
(60) 𝑠 = 4𝑅 sin
𝑐𝑎
2
=√
𝐶
, cos = √
2
2
𝑠
2∆
𝐵
𝑎𝑏
𝐴
2
∆
𝑎𝑐
(𝑠−𝑎)(𝑠−𝑏)
2
2
1
2 sin 𝐴
2
, tan
1
𝑎
𝑠(𝑠−𝑏)
𝐶
, sin = √
𝐶
(57) ∆= 𝑏𝑐 sin 𝐴 , ∆=
(58) 𝑅 =
𝑐𝑎
𝐵
, sin 2𝐵 =
𝑎−𝑏
(𝑠−𝑐)(𝑠−𝑎)
2
, cos = √
(𝑠−𝑏)(𝑠−𝑐)
2∆
𝐵
, sin = √
𝐴
2
1
𝑐𝑎 sin 𝐵, ∆= 𝑎𝑏 sin 𝐶
2
, 𝑅=
2 sin 𝐶
, 𝑅=
𝑎𝑏𝑐
4∆
𝐵
𝐶
2
2
, 𝑟 = (𝑠 − 𝑏 ) tan , 𝑟 = (𝑠 − 𝑐) tan
𝐵
𝐶
2
2
sin sin
𝑐