Solve cosA+sinA=sqrt{2}co

$$\cos A + \sin A = \sqrt { 2 } c o$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=-\frac{\sqrt{2}\left(\left(-1+i\right)e^{iA}+\left(-1-i\right)e^{-iA}\right)}{4o}\text{, }&o\neq 0\\c\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }A=\pi n_{1}+\frac{3\pi }{4}\text{ and }o=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{2}co=\cos(A)+\sin(A)$$

The equation is in standard form.

$$\sqrt{2}oc=\sin(A)+\cos(A)$$

Divide both sides by \sqrt{2}o.

$$\frac{\sqrt{2}oc}{\sqrt{2}o}=\frac{\sin(A)+\cos(A)}{\sqrt{2}o}$$

Dividing by \sqrt{2}o undoes the multiplication by \sqrt{2}o.

$$c=\frac{\sin(A)+\cos(A)}{\sqrt{2}o}$$

Divide \cos(A)+\sin(A) by \sqrt{2}o.

$$c=\frac{\sin(\frac{4A+\pi }{4})}{o}$$

Solve for c

$$\left\{\begin{matrix}c=\frac{\sin(\frac{4A+\pi }{4})}{o}\text{, }&o\neq 0\\c\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }A=\pi n_{1}+\frac{3\pi }{4}\text{ and }o=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{2}co=\cos(A)+\sin(A)$$

The equation is in standard form.

$$\sqrt{2}oc=\sin(A)+\cos(A)$$

Divide both sides by \sqrt{2}o.

$$\frac{\sqrt{2}oc}{\sqrt{2}o}=\frac{\sin(A)+\cos(A)}{\sqrt{2}o}$$

Dividing by \sqrt{2}o undoes the multiplication by \sqrt{2}o.

$$c=\frac{\sin(A)+\cos(A)}{\sqrt{2}o}$$

Divide \cos(A)+\sin(A) by \sqrt{2}o.

$$c=\frac{\sin(\frac{4A+\pi }{4})}{o}$$

Solve for A (complex solution)

$$A=-i\ln(\left(\frac{1}{2}+\frac{1}{2}i\right)\sqrt{\left(co\right)^{2}-1}+\left(\frac{1}{2}+\frac{1}{2}i\right)co)+2\pi n_{1}-\frac{\ln(2)i}{2}\text{, }n_{1}\in \mathrm{Z}$$

$$A=-i\ln(\left(-1-i\right)\sqrt{\left(co\right)^{2}-1}+\left(1+i\right)co)+2\pi n_{2}+\frac{\ln(2)i}{2}\text{, }n_{2}\in \mathrm{Z}$$