Compound 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 (see Pythagorean identities) 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).
Post by b***@hotmail.comI am in the roofing business, and for the life of me I cannot figure
out how to calculate the compound angle created by one roof meeting
another roof. This would be handy when we fabricate our valley
flashings. At this moment, we are stuck on how to bend our valley
flashings for a 16:12 pitch roof.
Thanks
Bob