z = a + bi
z = 4 + 4i
r² = a² + b²
r² = (4)² + (4)²
r² = 16 + 16
r² = 32
r = 4√(2)
r = 4(1.414)
r = 5.656
[tex]cos\theta = \frac{a}{r}[/tex]
[tex]cos\theta = \frac{4}{4\sqrt{2}}[/tex]
[tex]cos\theta = \frac{4}{4\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}}[/tex]
[tex]cos\theta = \frac{4\sqrt{2}}{4\sqrt{4}}[/tex]
[tex]cos\theta = \frac{4\sqrt{2}}{4(2)}[/tex]
[tex]cos\theta = \frac{4\sqrt{2}}{8}[/tex]
[tex]cos\theta = \frac{\sqrt{2}}{2}[/tex]
[tex]2(cos\theta) = 2(\frac{\sqrt{2}}{2})[/tex]
[tex]2cos\theta = \sqrt{2}[/tex]
[tex]2cos\theta = 1.414[/tex]
[tex]sin\theta = \frac{b}{r}[/tex]
[tex]sin\theta = \frac{4}{4\sqrt{2}}[/tex]
[tex]sin\theta = \frac{4}{4\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}}[/tex]
[tex]sin\theta = \frac{4\sqrt{2}}{4\sqrt{4}}[/tex]
[tex]sin\theta = \frac{4\sqrt{2}}{4(2)}[/tex]
[tex]sin\theta = \frac{4\sqrt{2}}{8}[/tex]
[tex]sin\theta = \frac{\sqrt{2}}{2}[/tex]
[tex]2(sin\theta) = 2(\frac{\sqrt{2}}{2})[/tex]
[tex]2sin\theta = \sqrt{2}[/tex]
[tex]2sin\theta = 1.414[/tex]
z = a + bi
z = rcosθ + (rsinθ)i
z = r(cosθ + i sinθ)
z = 4 + 4i
z = 5.656cosθ + (5.656sinθ)i
z = 5.656(cosθ + i sinθ)
z = 5.656(cos45 + i sin45)
[tex]\theta = tan^{-1}\frac{b}{a}[/tex]
[tex]\theta = tan^{-1}\frac{4}{4}[/tex]
[tex]\theta = tan^{-1}(1)[/tex]
[tex]\theta = 45[/tex]
The polar form of 4 + 4i is approximately equal to 5.656(cos45 + i sin45).
