Addition Formulae
sin(A + B) ≡ sinAcosB + cosAsinB
sin(A - B) ≡ sinAcosB - cosAsinB
cos(A + B) ≡ cosAcosB - sinAsinB
cos(A - B) ≡ cosAcosB + sinAsinB
tan(A + B) ≡ tanA + tanB
1 - tanAtanB
tan(A - B) ≡ tanA - tanB
1 + tanAtanB
These are given in combined form
sin(A - B) ≡ sinAcosB - cosAsinB
cos(A + B) ≡ cosAcosB - sinAsinB
cos(A - B) ≡ cosAcosB + sinAsinB
tan(A + B) ≡ tanA + tanB
1 - tanAtanB
tan(A - B) ≡ tanA - tanB
1 + tanAtanB
These are given in combined form
Double angle formulae
These aren't given but can be worked out as 2A = A + A
Rewriting acosx + bsinx as one function
acosx + bsinx can be written in any of the forms Rsin(x ± α) or Rcos(x ± α)
When R > 0 and 0° < α < 90°
Lets take an example:
4cosx + 3sinx ≡ Rcos(x + α)
Expand the right hand side to get using the addition formula:
4cosx + 3sinx ≡ Rcosxcosα - Rsinxsinα
Compare the coefficients of cosx and sinx:
4cosx + 3sinx ≡ Rcosxcosα - Rsinxsinα
4 = Rcosα
3 = Rsinα
We know that sinα/cosα = tanα, so:
3R = tanα
4R
R cancels out:
3 = tanα
4
This means α = 36.9°
3 = tanα means that we can create a right-angled triangle
4
When R > 0 and 0° < α < 90°
Lets take an example:
4cosx + 3sinx ≡ Rcos(x + α)
Expand the right hand side to get using the addition formula:
4cosx + 3sinx ≡ Rcosxcosα - Rsinxsinα
Compare the coefficients of cosx and sinx:
4cosx + 3sinx ≡ Rcosxcosα - Rsinxsinα
4 = Rcosα
3 = Rsinα
We know that sinα/cosα = tanα, so:
3R = tanα
4R
R cancels out:
3 = tanα
4
This means α = 36.9°
3 = tanα means that we can create a right-angled triangle
4
Overall this means that 4cosx + 3sinx = 5cos(x + 36.9)
This method can be used to solve equations and/or find the minimum or maximum value of x
Factor Formulae
To work out these formulae you need to add the two identities containing either cos or sin
E.g.
sin(A + B) + sin(A - B) = 2sinAcosB
A + B = P and A - B = Q
Through simultaneous equations you get what A and B are in P and Q
A = P + Q
2
B = P - Q
2
Giving you
2sin(P + Q)cos(P - Q)
2 2
E.g.
sin(A + B) + sin(A - B) = 2sinAcosB
A + B = P and A - B = Q
Through simultaneous equations you get what A and B are in P and Q
A = P + Q
2
B = P - Q
2
Giving you
2sin(P + Q)cos(P - Q)
2 2