# 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):

r^{2}cos^{2}a + r^{2}sin^{2}a = a^{2} + b^{2}

hence r^{2} = a^{2} + b^{2} (since cos^{2}a + sin^{2}a = 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 = cos^{2}A - sin^{2}A

Replacing cos^{2}A by 1 - sin^{2}A in the above formula gives:

cos2A = 1 - 2sin^{2}A

Replacing sin^{2}A by 1 - cos^{2}A gives:

cos2A = 2cos^{2}A - 1

It can also be shown that:

tan2A = __ 2tanA __

1 - tan^{2}A

**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).