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
\(cos\theta = \frac{a}{r}\)
\(cos\theta = \frac{4}{4\sqrt{2}}\)
\(cos\theta = \frac{4}{4\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}}\)
\(cos\theta = \frac{4\sqrt{2}}{4\sqrt{4}}\)
\(cos\theta = \frac{4\sqrt{2}}{4(2)}\)
\(cos\theta = \frac{4\sqrt{2}}{8}\)
\(cos\theta = \frac{\sqrt{2}}{2}\)
\(2(cos\theta) = 2(\frac{\sqrt{2}}{2})\)
\(2cos\theta = \sqrt{2}\)
\(2cos\theta = 1.414\)
\(sin\theta = \frac{b}{r}\)
\(sin\theta = \frac{4}{4\sqrt{2}}\)
\(sin\theta = \frac{4}{4\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}}\)
\(sin\theta = \frac{4\sqrt{2}}{4\sqrt{4}}\)
\(sin\theta = \frac{4\sqrt{2}}{4(2)}\)
\(sin\theta = \frac{4\sqrt{2}}{8}\)
\(sin\theta = \frac{\sqrt{2}}{2}\)
\(2(sin\theta) = 2(\frac{\sqrt{2}}{2})\)
\(2sin\theta = \sqrt{2}\)
\(2sin\theta = 1.414\)
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)
\(\theta = tan^{-1}\frac{b}{a}\)
\(\theta = tan^{-1}\frac{4}{4}\)
\(\theta = tan^{-1}(1)\)
\(\theta = 45\)
The polar form of 4 + 4i is approximately equal to 5.656(cos45 + i sin45).
