Solve sinθ-cosθ=4/5w

$$\sin \theta - \cos \theta = \frac { 4 } { 5 } w$$

Solve for w (complex solution)

$$w=\left(-\frac{5}{8}-\frac{5}{8}i\right)e^{i\theta }+\left(-\frac{5}{8}+\frac{5}{8}i\right)e^{-i\theta }$$

Steps for Solving Linear Equation

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

$$\frac{4}{5}w=\sin(\theta )-\cos(\theta )$$

Divide both sides of the equation by \frac{4}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{5}w}{\frac{4}{5}}=\frac{\sin(\theta )-\cos(\theta )}{\frac{4}{5}}$$

Dividing by \frac{4}{5} undoes the multiplication by \frac{4}{5}.

$$w=\frac{\sin(\theta )-\cos(\theta )}{\frac{4}{5}}$$

Divide \sin(\theta )-\cos(\theta ) by \frac{4}{5} by multiplying \sin(\theta )-\cos(\theta ) by the reciprocal of \frac{4}{5}.

$$w=\frac{5\left(\sin(\theta )-\cos(\theta )\right)}{4}$$

Solve for w

$$w=\frac{5\left(\sin(\theta )-\cos(\theta )\right)}{4}$$

Steps for Solving Linear Equation

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

$$\frac{4}{5}w=\sin(\theta )-\cos(\theta )$$

Divide both sides of the equation by \frac{4}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{5}w}{\frac{4}{5}}=\frac{\sin(\theta )-\cos(\theta )}{\frac{4}{5}}$$

Dividing by \frac{4}{5} undoes the multiplication by \frac{4}{5}.

$$w=\frac{\sin(\theta )-\cos(\theta )}{\frac{4}{5}}$$

Divide \sin(\theta )-\cos(\theta ) by \frac{4}{5} by multiplying \sin(\theta )-\cos(\theta ) by the reciprocal of \frac{4}{5}.

$$w=\frac{5\left(\sin(\theta )-\cos(\theta )\right)}{4}$$

Solve for θ (complex solution)

$$\theta =-i\ln(\left(-\frac{1}{10}+\frac{1}{10}i\right)\sqrt{16w^{2}-50}+\left(-\frac{2}{5}+\frac{2}{5}i\right)w)+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =-i\ln(\left(\frac{1}{10}-\frac{1}{10}i\right)\sqrt{16w^{2}-50}+\left(-\frac{2}{5}+\frac{2}{5}i\right)w)+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}$$