Trig functions of sums of angles
F = A cos β
C = F sin α
γ=α
D = G sin γ = G sin α
B = G cos γ = G cos α
G = A sin β
A
so
B+C
A
A sin β cos α + A cos β sin α
=
A
= cos α sin β + sin α cos β.
sin(α + β) =
and
β
F cos α − D
E
=
A
A
A cos β cos α − A sin β sin α
=
A
= cos α cos β − sin α sin β.
γ
B
D
F
α E
C
cos(α + β) =
In summary:
sin(α + β) = cos α sin β + sin α cos β,
cos(α + β) = cos α cos β − sin α sin β.
(1)
(2)
From these equations we can derive a number of other useful equations.
If we set α = β, we see that
sin(2α) = 2 cos α sin α
cos(2α) = cos2 α − sin2 α
= 1 − 2 sin2 θ
= 2 cos2 α − 1,
1
G
(3)
(4)
(5)
(6)
where we made use of cos2 α+sin2 α = 1 in deriving the alternate expressions
in the last equation. This last equation can be used to get an expression for
the sine or cosine of θ/2. By substituting α = θ/2, we get
s
sin(θ/2) =
1 − cos(θ)
,
2
(7)
1 + cos(θ)
.
2
(8)
s
cos(θ/2) =
In both cases the sign needs to be checked, because the square root could
take on either sign.
A+B
A−B
Finally, if we substitute α =
,β=
, into (1) we have
2
2
sin
A+B A−B
+
2
2
=
A+B
A−B
A+B
A−B
sin A = cos
sin
+ sin
cos
,
2
2
2
2
while if we substitute β =
A+B B−A
sin
+
2
2
B−A
instead we get
2
=
A+B
A−B
A+B
A−B
sin
+ sin
cos
,
sin B = − cos
2
2
2
2
Adding these two equations gives us
sin A + sin B = 2 sin
A+B
A−B
cos
.
2
2
(9)
Similarly, if we make the same substitution into (2), we have
cos A + cos B = cos(α + β) + cos(α − β)
= cos α cos β + sin α sin β + cos α cos β − sin α sin β = 2 cos α cos β
A+B
A−B
= 2 cos
cos
.
2
2
2