Sum of Sine and Cosine: Expressing Any Sum as C sin(α+ϕ)

[Dank2]

Homework Statement

Show that any sum:
Asin(α) + Bcos(α)

can be written as : C sin(α+ϕ)
2. Homework Equations

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.
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[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))

 

[SammyS]

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))

One backwards parenthesis .

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

Square each equation, then add them.

 

[Dank2]

One backwards parenthesis .

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

Square each equation, then add them.

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

 

[Dank2]

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

Then we got Csin(α + arctan(B/A)).

Thanks

 

[SammyS]

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

Then we got Csin(α + arctan(B/A)).

Thanks

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.