posted by Nathaniel. B .
One of the "think outside the box" problems in my math work is really challenging me.
I need to write and equation expressing:
8 cos theta + 15 sin theta as a single sinusoid
I tried graphing to no avail and there is nothing in my textbook about turning an equation like that into a sinusoidal equation.
Do a search for "sum of sin cosine phase shift" and at 200clicks there's a good write-up.
In summary, with the values of r and θ given by
r = sqrt(a2 + b2 + 2ab cos(A-B))
θ = atan2(a cos A + b cos B, a sin A + b sin B),
the following expressions are equivalent:
y = a cos(x-A) + b cos(x-B)
y = r cos(x-θ)
In your case, that means that y = 17cos(theta-62) is the wave you want.
Recall that sin theta = cos(90-theta)