# Homework Help: Sum of sine and cosine

1. Sep 22, 2016

### Dank2

1. The problem statement, all variables and given/known data
Show that any sum:
Asin(α) + Bcos(α)

can be written as : C sin(α+ϕ)
2. Relevant equations

3. The attempt at a solution
i can express cos(a) as as sin(90-a), and then try to use the formula that adds sines, but it gives the form of cos*sin.

2. Sep 22, 2016

### Dank2

After expressing C sin(α+ϕ), i managed to get the following:
A = C*cos(phi), B = C*sin(phi),

C = (A+B)/ (Cos(phi) + Sin(phi))

Last edited: Sep 22, 2016
3. Sep 22, 2016

### SammyS

Staff Emeritus
One backwards parenthesis .

Once you get A = C⋅cos(ϕ) and B = C⋅sin(ϕ),

Square each equation, then add them.

Last edited: Sep 22, 2016
4. Sep 22, 2016

### Dank2

A^2+B^2 = C2cos2phi + C2sin2phi = C2
C = sqrt(A2+B2)

5. Sep 22, 2016

### Dank2

then i can devide B/A = tan (Phi)
and then arctan B/A = Phi

Then we got Csin(α + arctan(B/A)).

Thanks

6. Sep 22, 2016

### SammyS

Staff Emeritus
You may need to be careful about the signs of A and B. (Well in this case, just the sign of A.)

If A is negative, then so is cos(ϕ), and that puts ϕ in the 2nd or 3rd quadrant.