I'll try to answer the question you wanted to ask (as indicated in the edited question) rather than the question as originally written,
whose answer you presumably already knew.
The question is,
if $y_1$ and $y_2$ are the following functions of $x$ and $t$,
\begin{align}
y_1 &= A_1 \sin(kx + \omega t + \delta_1) \\
y_2 &= A_2 \sin(kx + \omega t + \delta_2),
\end{align}
how do we find a definition of $y_1+y_2$ as a function of $x$ and $t$
in the form
$$
y_1 + y_2 = A_3 \sin(kx + \omega t + \delta_3)?
$$
To keep the formulas a little more manageable, let $u = kx + \omega t$.
Using the angle-sum formula for sine,
\begin{align}
\sin(kx + \omega t + \delta_1)
&= \sin\delta_1 \cos u + \cos\delta_1 \sin u, \\
\sin(kx + \omega t + \delta_2)
&= \sin\delta_2 \cos u + \cos\delta_2 \sin u.
\end{align}
Therefore
\begin{align}
y_1 + y_2 &= A_1(\sin\delta_1 \cos u + \cos\delta_1 \sin u)
+ A_2(\sin\delta_2 \cos u + \cos\delta_2 \sin u) \\
&= (A_1\sin\delta_1 + A_2\sin\delta_2)\cos u
+ (A_1\cos\delta_1 + A_2\cos\delta_2)\sin u.
\end{align}
So if $A = A_1\sin\delta_1 + A_2\sin\delta_2$
and $B = A_1\cos\delta_1 + A_2\cos\delta_2$,
we're looking for $A_3$ and $\delta_3$ that will generally satisfy
$$
A_3 \sin(u + \delta_3) = A \cos u + B \sin u.
$$
Working from this answer to a related question,
we want $A_3$ and $\delta_3$ such that $(A_3)^2 = A^2 + B^2$ and
$\tan\delta_3 = \frac AB$.
Recognizing that once we find a suitable $\delta_3$,
we can always add or subtract any whole multiple of $2\pi$
without changing the truth of any of our equations,
the solution is something of the form
\begin{gather}
A_3 = \pm\sqrt{A^2 + B^2}, \\
\delta_3 = \arctan \frac AB \quad\text{or}\quad
\delta_3 = \pi + \arctan \frac AB.
\end{gather}
But since
$A_3 \sin\left(u + \left(\pi + \arctan \frac AB\right)\right)
= -A_3 \sin\left(u + \left(\arctan \frac AB\right)\right)$,
we really have only two solutions to consider, and might as well
set $A_3$ to the positive square root of $A^2 + B^2$.
And then we merely need consider what happens when $u=0$,
that is, check whether $A_1 \sin(\delta_1) + A_2 \sin(\delta_2)$
is positive or negative, to decide whether $\sin(\delta_3)$
must be positive or negative and therefore whether to set
$\delta_3 = \arctan \frac AB$ or $\delta_3 = \pi + \arctan \frac AB$.
Alternatively, as one of my physics professors, Ken Lane, liked to say,
"let's complexify" the problem. We know that in general,
$$
e^{i\theta} = \cos \theta + i \sin \theta.
$$
If we take the real and imaginary parts of this function, we have two
sinusoidal functions $\frac\pi2$ radians out of phase.
As I recall, Prof. Lane liked to take the real part of this function as
the "wave", which is convenient if we want a phase-shifted cosine, but
you're looking for a phase-shifted sine, so it will be more convenient
to take the imaginary part.
The imaginary part of $e^{i\theta}$ is
$\newcommand{\Im}{\mathop{\mathrm{Im}}}
\Im\left(e^{i\theta}\right) = \sin\theta$,
and
$$
\Im\left(A_k e^{i(kx + \omega t + \delta_k)}\right)
= A_k \sin(kx + \omega t + \delta_k).
$$
So we can rewrite $y_1$ and $y_2$ as the imaginary parts of
\begin{align}
z_1 &= A_1 e^{i(kx + \omega t + \delta_1)}, \\
z_2 &= A_2 e^{i(kx + \omega t + \delta_2)}.
\end{align}
Adding the two functions together,
\begin{align}
z_1 + z_2 &= A_1 e^{i(kx + \omega t + \delta_1)}
+ A_2 e^{i(kx + \omega t + \delta_2)} \\
&= \left(A_1 e^{i\delta_1} + A_2 e^{i\delta_2}\right)
e^{i(kx + \omega t)} \\
&= (A_1 \cos\delta_1 + A_2 \cos\delta_2
+ i (A_1 \sin\delta_1 + A_2 \sin\delta_2))
e^{i(kx + \omega t)}
\end{align}
Let $A = A_1 \sin\delta_1 + A_2 \sin\delta_2$ and
$B = A_1 \cos\delta_1 + A_2 \cos\delta_2$; then we need merely
write the complex number $B + iA$ in the form $A_3 e^{i\delta_3}$,
which we do by setting $A_3 = \sqrt{A^2 + B^2}$
and either $\delta_3 = \arctan\frac AB$ (if $A$ is positive)
or $\delta_3 = \pi + \arctan\frac AB$ (if $A$ is negative).
We then have
$$
z_1 + z_2 = \left( A_3 e^{i \delta_3} \right) e^{i(kx + \omega t)}
= A_3 e^{i(kx + \omega t + \delta_3)}
$$
and
$$
y_1 + y_2 = \Im\left( z_1 + z_2 \right)
= A_3 \sin(kx + \omega t + \delta_3).
$$
