Why does $A\sin{k(x+c)}=a\sin{kx}+b\cos{kx}$ imply that $A=\sqrt{a^2+b^2}$ and $\tan{c}=-b/a$?

Here's my picture-proof of the identity, with $k=1$ and @MarkBennet's suggestion to remove the negative sign from the tangent:

$$p \sin(\bullet) + q \cos(\bullet) = r \sin(\bullet +\circ ), \quad\text{where}\quad r = \sqrt{p^2+q^2} \quad\text{and}\quad \tan(\circ) = \frac{q}{p}$$

I walk through the creation of the diagram in this answer.

You might want $\sin (kx+c)$ rather than $\sin k(x+c)$, and lose a minus sign on the $\tan$ expression, but the shape of the argument is as follows:

We start by noting that $A\sin (P+Q)=A\sin P\cos Q+A\cos P\sin Q$, for any values of $P,Q$ and we choose to use $P=kx$ and $Q=kc$

Now equating coefficients on the right-hand side we need $a=A\cos Q$ and $b=A\sin Q$ whence $a^2+b^2=A^2(\sin^2 Q+\cos^2 Q)=A^2$, and we also have $\tan Q=\cfrac ba$

This would give you what you wanted if you had $Q=c$ rather than $Q=kc$ and changed the sign - which is why I suggested checking the question carefully.