Trigonometric Formulas

Trigonometric Formulas
Half Angle Identities
s
sin( =2) =
s
cos( =2) =
1
cos( )
2
1 + cos( )
2
v
u
u1
t
cos( )
1 + cos( )
tan( =2) =
Power Reducing Identities
1
cos(2u)
2
1
+
cos(2u)
cos2 (u) =
2
1
cos(2u)
tan2 (u) =
1 + cos(2u)
sin2 (u) =
Product to Sum Formulas
1
[sin(u + v) + sin(u v)]
2
1
cos u sin v =
[sin(u + v) sin(u v)]
2
1
cos u cos v =
[cos(u + v) + cos(u v)]
2
1
sin u sin v =
[cos(u + v) cos(u v)]
2
sin u cos v =
Sum to Product Formulas
Angle
0
0
cos
0
1
tan
0
sin
30
6
1
p2
3
2
p
3
3
45
p4
2
p2
2
2
1
60
p3
3
2
1
2
p
3
sin
+ sin
sin
sin
cos
+ cos
cos
cos
90
2
1
0
+1
+
) cos(
)]
2
2
+
= 2[cos(
) sin(
)]
2
2
+
= 2[cos(
) cos(
)]
2
2
+
=
2[sin(
) sin(
)]
2
2
= 2[sin(
120
135
150
2
p3
3
2
1
2
3
p4
2
2
p
2
2
5
6
1
2
p
p
3
1
2p
180
0
1
3
3
0
3
1
210
225
240
270
300
315
7
6
1
2
p
3
2
p
5
4
p
4
3
p
3
2
5
3
p
7
4
p
3
3
2
p
2
1
2
2
3
2
1
2
p
3
1
0
1
2
1
2
3
p
3
2
p2
2
2
1
330 360
11
2
6
1
p2
3
2p
3
3
0
1
0
Quotient identities
sin(
cos(
cos(
cot( ) =
sin(
tan( ) =
)
)
)
)
Reciprocal identities
1
cos( )
1
csc( ) =
sin( )
1
cot( ) =
tan( )
sec( ) =
Pythagorean identities
sin2 ( ) + cos2 ( ) = 1
1 + tan2 ( ) = sec2 ( )
1 + cot2 ( ) = csc2 ( )
Sum and Di erence formulas
sin(u + v)
sin(u v)
cos(u + v)
cos(u v)
=
=
=
=
sin(u) cos(v) + cos(u) sin(v)
sin(u) cos(v) cos(u) sin(v)
cos(u) cos(v) sin(u) sin(v)
cos(u) cos(v) sin(u) sin(v)
tan(u) + tan(v)
tan(u + v) =
1 tan(u) tan(v)
tan(u) tan(v)
tan(u v) =
1 + tan(u) tan(v)
Double angle formulas
sin(2 ) =
cos(2 ) =
=
=
2 sin( ) cos( )
cos2 ( ) sin2 ( )
2 cos2 ( ) 1
1 2 sin2 ( )
2 tan( )
tan(2 ) =
1 tan2 ( )
2