Ksin(x+a)

4sinx+5cosx in the form ksin(x+a)

$\boxed{~ \begin{array}{rcll} A\cdot \sin{(x)}+ B \cdot \cos{(x)} &=& k\cdot \sin{(x+a)} \\ && k = \sqrt{A^2+B^2} \qquad a = \tan^{-1}{ \left(\frac{B}{A} \right)} \end{array} ~}$

$\begin{array}{rcll} A\cdot \sin{(x)}+ B \cdot \cos{(x)} &=& k\cdot \sin{(x+a)} \qquad A = 4 \qquad B = 5 \\ \qquad && k = \sqrt{4^2+5^2} \qquad a = \tan^{-1}{ \left(\frac{5}{4} \right)} \\ \qquad && k = \sqrt{16+25} \qquad a = \tan^{-1}{ (1.25 )} \\ \qquad && k = \sqrt{41} \qquad \qquad a = 51.3401917459^\circ\\\\ 4\cdot \sin{(x)}+ 5 \cdot \cos{(x)} &=& \sqrt{41}\cdot \sin{(x+51.3401917459^\circ )} \\ 4\cdot \sin{(x)}+ 5 \cdot \cos{(x)} &=& 6.40312423743\cdot \sin{(x+51.3401917459^\circ )} \\ \end{array}$