Q.Two parallel S.H.M.s represented by $x_1=5\sin(4\pi t+\pi/3)$ cm and $x_2=3\sin(4\pi t+\pi/4)$ cm are superposed on a particle. Determine the amplitude and epoch of the resultant S.H.M.

[!TLDR] $A_1=5$, $A_2=3$, $\phi_1-\phi_2=\pi/12=15°$. $R^2=25+9+30\cos15°\approx62.98\Rightarrow R\approx7.936$ cm. $\tan\delta\approx1.396\Rightarrow\delta\ …

Given $x_1=5\sin(4\pi t+\pi/3)$ and $x_2=3\sin(4\pi t+\pi/4)$, so $A_1=5$, $\phi_1=\pi/3=60°$; $A_2=3$, $\phi_2=\pi/4=45°$. The resultant amplitude (Eq. 5.19): $R^2=A_1^2+A_2^2+2A_1A_2\cos(\phi_1-\phi_2)=25+9+2(5)(3)\cos(15°)=34+30(0.9659)=34+28.977=62.977$ so $R=\sqrt{62.977}\approx7.936$ cm. The resultant initial phase $\delta$ (Eq. 5.20): $$\tan\delta=\frac{A_1\sin\phi_1+A_2\sin\phi_2}{A_1\cos\phi_1+A_2\cos\phi_2}=\frac{5\sin60°+3\sin45°}{5 …

Read off $A_1,A_2,\phi_1,\phi_2$, apply Eq. (5.19) for R and Eq. (5.20) for tan(delta), the …

Using $(\phi_1-\phi_2)$ in degrees inside a cosine expecting radians (or vice versa) when computing $R^ …