R-form

In $3\sin\theta+4\cos\theta$, both ratios have the same input. Expand $R\sin(\theta+\alpha)$ and choose the constants $R,\alpha$ so its coefficients match.

Square and add the last two equations. Since $\cos^2\alpha+\sin^2\alpha=1$, this gives $R^2=3^2+4^2=25$. Take $R>0$, so $R=5$. Divide the second equation by the first to find $\tan\alpha=4/3$.

Both sine and cosine of $\alpha$ are positive, so choose $0^\circ<\alpha<90^\circ$. Thus $\alpha=\tan^{-1}(4/3)\approx53.1^\circ$.

This is an exact identity with the exact $\alpha$. Using $53.1^\circ$ instead makes it approximate. Store the full calculator value for later working. R-form is derived this way; it is not a separate formula supplied in MF19.

Amplitude is the distance from the midline to a peak. Here it is $R$, not $3+4$: sine and cosine do not reach 1 at the same angle. The phase shift moves the wave horizontally. In $\sin(\theta+\alpha)$, a positive $\alpha$ moves it left.

Change a coefficient and compare the two component waves with their sum. Try a negative coefficient, then make both zero.

This method needs the same input angle in both terms. It works with $a\sin(2x)+b\cos(2x)$, treating $2x$ as one input. It does not directly combine $a\sin x+b\cos2x$.