## Trigonometry (11th Edition) Clone

The four possible value for $x$ are: $2~(cos~45^{\circ}+i~sin~45^{\circ})$ $2~(cos~135^{\circ}+i~sin~135^{\circ})$ $2~(cos~225^{\circ}+i~sin~225^{\circ})$ $2~(cos~315^{\circ}+i~sin~315^{\circ})$
$x^4+16 = 0$ $x^4 = -16$ $x = (-16)^{1/4}$ $x = [16(-1+0~i)]^{1/4}$ Let $z = 16(-1 + 0~i)$ $z = 16(cos~180^{\circ}+i~sin~180^{\circ})$ $r = 16$ and $\theta = 180^{\circ}$ We can use this equation to find the fourth roots: $z^{1/n} = r^{1/n}~[cos(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})+i~sin(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})]$, where $k \in \{0, 1, 2,...,n-1\}$ When k = 0: $z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})]$ $z^{1/4} = 2~(cos~45^{\circ}+i~sin~45^{\circ})$ When k = 1: $z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})]$ $z^{1/4} = 2~(cos~135^{\circ}+i~sin~135^{\circ})$ When k = 2: $z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})]$ $z^{1/4} = 2~(cos~225^{\circ}+i~sin~225^{\circ})$ When k = 3: $z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})]$ $z^{1/4} = 2~(cos~315^{\circ}+i~sin~315^{\circ})$