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).