Trig Formulas

TRIGONOMETRY FORMULAS
Angle Addition Formulas
Difference Formulas
sin(α + β ) = sinα cos β + cosαsin β
sin(α − β ) = sinα cos β − cosαsin β
cos(α + β ) = cosα cos β − sinα sin β
cos(α − β ) = cosαcos β + sinα sin β
tan (α + β ) =
tan a + tan β
1 − tan α tan β
tan (α − β ) =
tan a − tan β
1 + tan α tan β
Half Angle Formulas
Double Angle Formulas
sin(2θ) = 2 sin θ cosθ
cos(2θ) = cos θ − sin θ
2
2
= 1 − 2 sin 2 θ
= 2 cos2 θ − 1
tan (2θ) =
2 tanθ
1 − tan 2 θ
 u
1− cosu
sin  = ±
 2
2
 u
1+ cosu
cos  = ±
 2
2
 u
1− cosu
tan  = ±
 2
1+ cosu
1 − cosu
=
sinu
sinu
=
1 + cosu
Product-to-Sum Formulas
 u + v   u − v
sin u + sin v = 2sin 
 cos

 2   2 
1
[cos( A − B ) − cos( A + B )]
2
1
cos A cosB = [cos( A − B ) + cos( A + B)]
2
1
sin A cos B = [sin( A + B ) + sin ( A − B )]
2
1
cos Asin B = [sin( A + B ) − sin ( A − B )]
2
sin A sin B sin C
=
=
a
b
c
Law of Cosines
SAS Form
c 2 = a 2 + b 2 − 2ab cosC
b 2 = c 2 + a 2 − 2ca cos B
a 2 = b 2 + c 2 − 2bc cos A
1 − cos(2θ )
2
1+ cos(2θ )
cos 2 θ =
2
1 − cos(2θ)
2
tan θ =
1 + cos(2θ)
sin 2 θ =
Sum-to-Product Formulas
sin Asin B =
Law of Sines
Power Reducing Formulas
 u + v  u − v
sin u − sin v = 2 cos
 sin

 2   2 
 u + v
 u − v
cosu + cosv = 2 cos
 cos

 2 
 2 
 u + v  u − v 
cosu − cosv = −2 sin 
 sin 

 2   2 
Law of Cosines
SSS Form
Area of a Triangle (SAS)
b +c − a
2 bc
c2 + a 2 − b2
cos B =
2ca
a 2 + b 2 − c2
cosC =
2ab
cos A =
2
2
2
=
absin C bcsin A casin B
=
=
2
2
2
Heron’s Formula (SSS)
s=
a+b +c
2
Area = s (s − a)(s − b )( s − c )