Compound Angle Formulae
This section covers compound angle formulae and double angle formulae.
sin(A + B) DOES NOT equal sinA + sinB. Instead, you must expand such expressions using the formulae below.
The following are important trigonometric relationships:
sin(A + B) = sinAcosB + cosAsinB
cos(A + B) = cosAcosB - sinAsinB
tan(A + B) = tanA + tanB
1 - tanAtanB
To find sin(A - B), cos(A - B) and tan(A - B), just change the + signs in the above identities to - signs and vice-versa:
sin(A - B) = sinAcosB - cosAsinB
cos(A - B) = cosAcosB + sinAsinB
tan(A - B) = tanA - tanB
1 + tanAtanB
rcos(q + a) form
When we have an expression in the form: acosq + bsinq, it is sometimes best to rewrite this in the form rcos(q + a), especially when solving trigonometric equations.
To calculate what r and a are, note that rcos(q + a) = r cosq cosa - r sinq sina = r cosa cosq - r sina sinq by the above identity.
So we need to set rcosa = a and -rsina = b to make this equal to acosq + bsinq .
So we have two equations:
rcosa = a (1)
rsina = -b (2)
We can find a by dividing (2) by (1):
sina/cosa = -b/a , hence tana = -b/a which we can solve.
We can find r by squaring and adding (1) and (2):
r2cos2a + r2sin2a = a2 + b2
hence r2 = a2 + b2 (since cos2a + sin2a = 1)
In a similar way, we can write expressions of the form acosq + bsinq as rsin(q + a).
Double Angle Formulae
sin(A + B) = sinAcosB + cosAsinB
Replacing B by A in the above formula becomes:
sin(2A) = sinAcosA + cosAsinA
so: sin2A = 2sinAcosA
similarly:
cos2A = cos2A - sin2A
Replacing cos2A by 1 - sin2A in the above formula gives:
cos2A = 1 - 2sin2A
Replacing sin2A by 1 - cos2A gives:
cos2A = 2cos2A - 1
It can also be shown that:
tan2A = 2tanA
1 - tan2A
Product to Sum Formulae
Sometimes it is useful to be able to write a product of trigonometric functions as a sum of simpler trigonometric functions (this might make integration easier, for example).
Now, cos(A + B) = cosAcosB - sinAsinB
and cos(A - B) = cosAcosB + sinAsinB
Adding these two:
cos(A + B) + cos(A - B) = 2cosAcosB
Subtracting one from the other:
cos(A - B) - cos(A + B) = 2sinAsinB
Similar formula can be obtained using the expansion of sin(A + B).
