Solve Z=3cos(5pit)+3jcos(5pit)

$$Z = 3 \cos ( 5 \pi t ) + 3 j \cos ( 5 \pi t )$$

Solve for j (complex solution)

$$\left\{\begin{matrix}j=-\frac{3\cos(5\pi t)-Z}{3\cos(5\pi t)}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }t=\frac{n_{1}}{5}+\frac{1}{10}\\j\in \mathrm{C}\text{, }&Z=3\cos(5\pi t)\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }t=\frac{2n_{1}+1}{10}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$3\cos(5\pi t)+3j\cos(5\pi t)=Z$$

Subtract 3\cos(5\pi t) from both sides.

$$3j\cos(5\pi t)=Z-3\cos(5\pi t)$$

The equation is in standard form.

$$3\cos(5\pi t)j=-3\cos(5\pi t)+Z$$

Divide both sides by 3\cos(5\pi t).

$$\frac{3\cos(5\pi t)j}{3\cos(5\pi t)}=\frac{-3\cos(5\pi t)+Z}{3\cos(5\pi t)}$$

Dividing by 3\cos(5\pi t) undoes the multiplication by 3\cos(5\pi t).

$$j=\frac{-3\cos(5\pi t)+Z}{3\cos(5\pi t)}$$

Divide Z-3\cos(5\pi t) by 3\cos(5\pi t).

$$j=\frac{\frac{Z}{\cos(5\pi t)}-3}{3}$$

Solve for Z

$$Z=3\left(j+1\right)\cos(5\pi t)$$