Product to Sum Formulas
s
i
n
(
u
)
s
i
n
(
v
)
=
1
2
[
c
o
s
(
u
−
v
)
−
c
o
s
(
u
+
v
)
]
{\displaystyle sin(u)sin(v)={\frac {1}{2}}[cos(u-v)-cos(u+v)]}
c
o
s
(
u
)
c
o
s
(
v
)
=
1
2
[
c
o
s
(
u
−
v
)
+
c
o
s
(
u
+
v
)
]
{\displaystyle cos(u)cos(v)={\frac {1}{2}}[cos(u-v)+cos(u+v)]}
s
i
n
(
u
)
c
o
s
(
v
)
=
1
2
[
s
i
n
(
u
+
v
)
+
s
i
n
(
u
−
v
)
]
{\displaystyle sin(u)cos(v)={\frac {1}{2}}[sin(u+v)+sin(u-v)]}
c
o
s
(
u
)
s
i
n
(
v
)
=
1
2
[
s
i
n
(
u
+
v
)
−
s
i
n
(
u
−
v
)
]
{\displaystyle cos(u)sin(v)={\frac {1}{2}}[sin(u+v)-sin(u-v)]}
Sum to Product Formula
s
i
n
(
u
)
+
s
i
n
(
v
)
=
2
s
i
n
(
u
+
v
2
)
c
o
s
(
u
−
v
2
)
{\displaystyle sin(u)+sin(v)=2sin({\frac {u+v}{2}})cos({\frac {u-v}{2}})}
s
i
n
(
u
)
−
s
i
n
(
v
)
=
2
c
o
s
(
u
+
v
2
)
s
i
n
(
u
−
v
2
)
{\displaystyle sin(u)-sin(v)=2cos({\frac {u+v}{2}})sin({\frac {u-v}{2}})}
c
o
s
(
u
)
+
c
o
s
(
v
)
=
2
c
o
s
(
u
+
v
2
)
c
o
s
(
u
−
v
2
)
{\displaystyle cos(u)+cos(v)=2cos({\frac {u+v}{2}})cos({\frac {u-v}{2}})}
c
o
s
(
u
)
−
c
o
s
(
v
)
=
−
2
s
i
n
(
u
+
v
2
)
s
i
n
(
u
−
v
2
)
{\displaystyle cos(u)-cos(v)=-2sin({\frac {u+v}{2}})sin({\frac {u-v}{2}})}
This material was adapted from the original CK-12 book that can be found here . This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License
![{\displaystyle sin(u)sin(v)={\frac {1}{2}}[cos(u-v)-cos(u+v)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49b9f060b7054ceeb498513300acc3896873419d)
![{\displaystyle cos(u)cos(v)={\frac {1}{2}}[cos(u-v)+cos(u+v)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b5c657f67a82c2a194619f642a34b5f3f7c666)
![{\displaystyle sin(u)cos(v)={\frac {1}{2}}[sin(u+v)+sin(u-v)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd3e38489f2f2318957665d6a57024d6393d7f6)
![{\displaystyle cos(u)sin(v)={\frac {1}{2}}[sin(u+v)-sin(u-v)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aeb7775ca872f16d7239754d52767bf0d10f005)
