# Converting from Polar form to Cartesian Form

I am just starting with complex numbers and vectors. The question is:

Convert the following to Cartesian form.

a) $8 \,\text{cis} \frac \pi4$

The formula given is:

$$z = x +yi = r\space(\cos\theta + i\sin\theta)$$

With $r=8$ and $\theta = \frac\pi4$, I did:

$$z=8\left(\cos\frac\pi4 + i \sin \frac\pi4\right)$$ $$z = 8(0.71 + i0.71)&$$z = 5.66 + i5.66$$The answer they give in the answers section is:$$z=4\sqrt2(1+i)$$I know this is the same answer just written approximately. Is someone able to take me through how you get to z=4\sqrt2(1+i) instead of z = 5.66 + i5.66? • \cos\dfrac\pi4=\dfrac1{\sqrt2} – lab bhattacharjee Feb 27 '18 at 4:05 • I know this is the same answer just written differently No, because \,4 \sqrt{2} \ne 5.66\,. – dxiv Feb 27 '18 at 4:13 • @dxiv Thank you. Should have written approximately. – Juan Pablo Feb 27 '18 at 4:15 • Also, while everyone is here, does anyone know why when I write fractions in Mathsjax they come out so small? And everyone elses are normal size. I am new to stack exchange and mathsjax – Juan Pablo Feb 27 '18 at 4:16 • @JuanPablo That was the point. When an exact result is readily available, just an approximation won't cut it. P.S. About fractions, try $$\frac{a}{b}$$ or \dfrac{a}{b} or \displaystyle\frac{a}{b}. See here for more MathJax tips. – dxiv Feb 27 '18 at 4:17 ## 1 Answer Knowing the \pi\over4 family:$$\cos \dfrac{\pi}{4} = \sin \dfrac{\pi}{4} = \dfrac{1}{\sqrt 2}$$Then applying this result to 8\,\text{cis} \frac{\pi}{4}:$$z=8\left(\dfrac{1}{\sqrt 2} + i\dfrac{1}{\sqrt 2}\right) = 8\cdot \dfrac{1}{\sqrt 2}(1+i)$$And then multiply:$$8\cdot \dfrac{1}{\sqrt 2} = \dfrac{8\sqrt 2}{2}=4\sqrt28\cdot \dfrac{1}{\sqrt 2}(1+i) = 4\sqrt 2(1+i)

• Ok, that makes sense, I was stuck at the very first bit. Much appreciated. – Juan Pablo Feb 27 '18 at 4:12
• @JuanPablo You've got to remember your unit circle :) – Andrew Li Feb 27 '18 at 4:13