Trigonometric Formulas:
Below are the list of all trigonometric functions formulas;
Basic Formulas:
sin (θ) = 1 csc(θ)
OR
csc (θ) = 1 sin (θ)
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cos (θ) = 1 sec(θ)
OR
sec (θ) = 1 cos (θ)
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tan (θ) = sin (θ) cos (θ)
OR
tan (θ) = 1 cot (θ)
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cot (θ) = cos (θ) sin (θ)
OR
cot (θ) = 1 tan (θ)
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sin2 (θ) + cos2 (θ) = 1
OR
sin2 (θ) = 1 - cos2 (θ)
OR
cos2 (θ) = 1 - sin2 (θ)
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1 + tan2 (θ) = sec2 (θ)
OR
tan2 (θ) = sec2 (θ) - 1
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1 + cot2 (θ) = csc2 (θ)
OR
cot2 (θ) = csc2 (θ) - 1
=====
csc (θ) = 1 sin (θ)
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sec (θ) = 1 cos (θ)
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NEGATIVE(-ve) ANGLE FORMULAS:
sin (-θ) = - sin (θ)
cos (-θ) = cos (θ)
tan (-θ) = - tan (θ)
cot (-θ) = - cot (θ)
sec (-θ) = sec (θ)
csc (-θ) = - csc (θ)
sin (-θ) = - sin (θ)
cos (-θ) = cos (θ)
tan (-θ) = - tan (θ)
cot (-θ) = - cot (θ)
sec (-θ) = sec (θ)
csc (-θ) = - csc (θ)
cos (-θ) = cos (θ)
tan (-θ) = - tan (θ)
cot (-θ) = - cot (θ)
sec (-θ) = sec (θ)
csc (-θ) = - csc (θ)
DOUBLE ANGLE FORMULAS:
sin (2θ) = 2 sin (θ) cos (θ)
OR
sin (2θ) = 2tan(θ)1 + tan2 (θ)
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cos (2θ) = cos2 (θ) - sin2 (θ)
OR
cos (2θ) = 2 cos2 (θ) - 1
OR
cos (2θ) = 1 - 2 sin2 (θ)
OR
cos (2θ) = 1 - tan2 (θ)1 + tan2 (θ)
= = = = =
tan (2θ) = 2 tan (θ)1- tan2 (θ)
sin (2θ) = 2 sin (θ) cos (θ)
OR
sin (2θ) = 2tan(θ)1 + tan2 (θ)
=====
cos (2θ) = cos2 (θ) - sin2 (θ)
OR
cos (2θ) = 2 cos2 (θ) - 1
OR
cos (2θ) = 1 - 2 sin2 (θ)
OR
cos (2θ) = 1 - tan2 (θ)1 + tan2 (θ)
= = = = =
tan (2θ) = 2 tan (θ)1- tan2 (θ)
HALF ANGLE FORMULAS:
sin ( α2 ) = ± √ 1- cos ( α )2
cos ( α2 ) = ± √ 1+ cos ( α )2
tan ( α2 ) = ± √ 1- cos ( α )1+ cos ( α )
sin ( α2 ) = ± √ 1- cos ( α )2
cos ( α2 ) = ± √ 1+ cos ( α )2
tan ( α2 ) = ± √ 1- cos ( α )1+ cos ( α )
FUNDAMENTAL LAWS:
sin ( α+ β )= sin ( α ) cos ( β )+ cos ( α ) sin ( β )
sin ( α- β )= sin ( α )cos ( β )- cos ( α ) sin (β )
cos ( α+ β )= cos ( α )cos ( β )- sin ( α ) sin (β )
cos ( α- β )= cos ( α )cos ( β )+ sin ( α ) sin (β )
tan ( α+ β )= tan ( α )+ tan ( β )1- tan ( α ) tan ( β )
tan ( α- β )= tan ( α )- tan ( β )1+ tan ( α ) tan ( β )
cot ( α+ β )= cot ( α) cot ( β )-1 cot ( α )+ cot ( β )
cot ( α- β )= cot ( α) cot ( β )+1 cot ( β )- cot ( α )
PRODUCT TO SUM FORMULAS:
sin ( α ) cos ( β )= 12 [ sin ( α+ β )+ sin (α- β ) ]
cos ( α ) sin ( β )= 12 [ sin ( α+ β )- sin (α- β ) ]
cos ( α ) cos ( β )= 12 [ cos ( α+ β )+ cos (α- β ) ]
sin ( α ) sin ( β )=- 12 [ cos ( α+ β )- cos (α- β ) ]
sin ( α ) cos ( β )= 12 [ sin ( α+ β )+ sin (α- β ) ]
cos ( α ) sin ( β )= 12 [ sin ( α+ β )- sin (α- β ) ]
cos ( α ) cos ( β )= 12 [ cos ( α+ β )+ cos (α- β ) ]
sin ( α ) sin ( β )=- 12 [ cos ( α+ β )- cos (α- β ) ]
SUM TO PRODUCT FORMULAS:
sin ( υ )+ sin ( ν )=2 sin( υ+ ν2 ) cos ( υ- ν2 )
sin ( υ )- sin ( ν )=2 cos ( υ+ ν2 ) sin ( υ- ν2 )
cos ( υ )+ cos ( ν )=2 cos ( υ+ ν2 ) cos ( υ- ν2 )
cos ( υ )- cos ( ν )=-2 sin ( υ+ ν2 ) sin ( υ- ν2 )
sin ( υ )+ sin ( ν )=2 sin( υ+ ν2 ) cos ( υ- ν2 )
sin ( υ )- sin ( ν )=2 cos ( υ+ ν2 ) sin ( υ- ν2 )
cos ( υ )+ cos ( ν )=2 cos ( υ+ ν2 ) cos ( υ- ν2 )
cos ( υ )- cos ( ν )=-2 sin ( υ+ ν2 ) sin ( υ- ν2 )
LAW OF SINE:
sin ( α )a= sin ( β )b= sin ( γ )c
OR
a sin ( α )= bsin ( β )= c sin ( γ )
OR
a : b : c = sin ( α ) : sin ( β ) : sin ( γ )
sin ( α )a= sin ( β )b= sin ( γ )c
OR
a sin ( α )= bsin ( β )= c sin ( γ )
OR
a : b : c = sin ( α ) : sin ( β ) : sin ( γ )
LAW OF COSINE:
cos ( α )= b2+c2-a22bc
OR
a2=b2+c2-2bc cos ( α )
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cos ( β )= a2+c2-b22ac
OR
b2=a2+c2-2ac cos ( β )
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cos ( γ )= a2+b2-c22ab
OR
c2=a2+b2-2ab cos ( γ )
cos ( α )= b2+c2-a22bc
OR
a2=b2+c2-2bc cos ( α )
=====
cos ( β )= a2+c2-b22ac
OR
b2=a2+c2-2ac cos ( β )
=====
cos ( γ )= a2+b2-c22ab
OR
c2=a2+b2-2ab cos ( γ )
HALF ANGLE FORMULAS:
sin ( α2 ) = √ (s-b) (s-c)bc
sin ( β2 ) = √ (s-a) (s-c)ac
sin ( γ2 ) = √ (s-a) (s-b)ab
cos ( α2 ) = √ s(s-a)bc
cos ( β2 ) = √ s(s-b)ac
cos ( γ2 ) = √ s(s-c)ab
tan ( α2 ) = √ (s-b) (s-c)s(s-a)
tan ( β2 ) = √ (s-a) (s-c)s(s-b)
tan ( γ2 ) = √ (s-a) (s-b)s(s-c)
sin ( α2 ) = √ (s-b) (s-c)bc
sin ( β2 ) = √ (s-a) (s-c)ac
sin ( γ2 ) = √ (s-a) (s-b)ab
cos ( α2 ) = √ s(s-a)bc
cos ( β2 ) = √ s(s-b)ac
cos ( γ2 ) = √ s(s-c)ab
tan ( α2 ) = √ (s-b) (s-c)s(s-a)
tan ( β2 ) = √ (s-a) (s-c)s(s-b)
tan ( γ2 ) = √ (s-a) (s-b)s(s-c)
AREA OF A TRIANGLE:
â–³=Area
â–³=Area
3 SIDES GIVEN:
△= √s(s-a)(s-b)(s-c)
s= a+b+c2
△= √s(s-a)(s-b)(s-c)
s= a+b+c2
2 SIDES INCLUDED ANGLE GIVEN:
△= bc sin ( α )2
△= ac sin ( β )2
△= ab sin ( γ )2
△= bc sin ( α )2
△= ac sin ( β )2
△= ab sin ( γ )2
1 SIDES INCLUDED 2 ANGLES GIVEN:
△= a2 sin ( β ) sin ( γ )2 sin ( α )
△= b2 sin ( α ) sin ( γ )2 sin ( β )
△= c2 sin ( α ) sin ( β )2 sin ( γ )
△= a2 sin ( β ) sin ( γ )2 sin ( α )
△= b2 sin ( α ) sin ( γ )2 sin ( β )
△= c2 sin ( α ) sin ( β )2 sin ( γ )
CIRCUM-RADIUS R IN TERMS OF THE MEASURE OF A THREE SIDES OF A TRIANGLE:
R = abc4â–³
R = abc4â–³
THE IN-RADIUS OF A TRIANGLE:
r= â–³s
r= â–³s