Solve {l}{sin(θ)+cos(θ)=1/2}{a=cos(θ)-sin(θ)}{b=a}{c=b}{text{Solvefor}dtext{where}}{d=c}

$$\left. \begin{array} { l } { \sin(\theta) + \cos(\theta) = \frac{1}{2} }\\ { a = \cos(\theta) - \sin(\theta) }\\ { b = a }\\ { c = b }\\ { \text{Solve for } d \text{ where} } \\ { d = c } \end{array} \right.$$

Solve for θ, a, b, c, d

$$\theta =2\pi n_{1}-i\ln(\frac{\sqrt{7}\left(1-i\right)+\left(1+i\right)}{4})\text{, }n_{1}\in \mathrm{Z}\text{, }a=-\frac{\sqrt{7}\left(-1-i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(1-i\right)+\left(1+i\right)\right)}\approx 1.322875656\text{, }b=-\frac{\sqrt{7}\left(-1-i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(1-i\right)+\left(1+i\right)\right)}\approx 1.322875656\text{, }c=-\frac{\sqrt{7}\left(-1-i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(1-i\right)+\left(1+i\right)\right)}\approx 1.322875656\text{, }d=-\frac{\sqrt{7}\left(-1-i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(1-i\right)+\left(1+i\right)\right)}\approx 1.322875656$$

$$\theta =2\pi n_{2}-i\ln(\frac{\sqrt{7}\left(-1+i\right)+\left(1+i\right)}{4})\text{, }n_{2}\in \mathrm{Z}\text{, }a=-\frac{\sqrt{7}\left(1+i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(-1+i\right)+\left(1+i\right)\right)}\approx -1.322875656\text{, }b=-\frac{\sqrt{7}\left(1+i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(-1+i\right)+\left(1+i\right)\right)}\approx -1.322875656\text{, }c=-\frac{\sqrt{7}\left(1+i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(-1+i\right)+\left(1+i\right)\right)}\approx -1.322875656\text{, }d=-\frac{\sqrt{7}\left(1+i\right)+\left(-7+7i\right)}{2\left(\sqrt{7}\left(-1+i\right)+\left(1+i\right)\right)}\approx -1.322875656$$