# How do you add (2+9i)+(5-7i) in trigonometric form?

May 21, 2018

$\left(2 + 9 i\right) + \left(5 - 7 i\right) = \sqrt{53} \left(\cos \left(0.279\right) - \sin \left(0.279\right) i\right)$

#### Explanation:

We add two complex numbers $a + b i$ and $c + \mathrm{di}$ as follows:

$\left(a + b i\right) + \left(c + \mathrm{di}\right) : = \left(a + c\right) + \left(b + d\right) i$

So $\left(2 + 9 i\right) + \left(5 - 7 i\right) = 7 - 2 i$. Now, to convert to trigonometric, we use the following identity:

$\left(a + b i\right) = r \left(\cos \left(\theta\right) + \sin \left(\theta\right) i\right)$

where $r = \sqrt{{a}^{2} + {b}^{2}}$ and $\theta$ satisifies $\cos \theta = \frac{a}{r} , \sin \theta = \frac{b}{r}$

So, for $7 - 2 i$, $r = \sqrt{49 + 4} = \sqrt{53}$ and $\cos \theta = 7 \text{/"sqrt53,sintheta=-2"/} \sqrt{53}$ and so $\theta = \arctan \left(- 2 \text{/} 7\right) \approx - 0.279$

Finally, $7 - 2 i = \sqrt{53} \left(\cos \left(- 0.279\right) + \sin \left(- 0.279\right) i\right) = \sqrt{53} \left(\cos \left(0.279\right) - \sin \left(0.279\right) i\right)$

Jul 9, 2018

color(violet)(=> 7 + 2 i

#### Explanation:

$z = a + b i = r \left(\cos \theta + i \sin \theta\right)$

$r = \sqrt{{a}^{2} + {b}^{2}} , \text{ } \theta = {\tan}^{-} 1 \left(\frac{b}{a}\right)$

${r}_{1} \left(\cos \left({\theta}_{1}\right) + i \sin \left({\theta}_{2}\right)\right) + {r}_{2} \left(\cos \left({\theta}_{2}\right) + i \sin \left({\theta}_{2}\right)\right) = {r}_{1} \cos \left({\theta}_{1}\right) + {r}_{2} \cos \left({\theta}_{2}\right) + i \left({r}_{1} \sin \left({\theta}_{1}\right) + {r}_{2} \sin \left({\theta}_{2}\right)\right)$

r_1=sqrt(2^2+ 9^2))=sqrt 85
${r}_{2} = \sqrt{{5}^{2} + - {7}^{2}} = \sqrt{74}$

${\theta}_{1} = {\tan}^{-} 1 \left(\frac{9}{2}\right) \approx {77.47}^{\circ} , \text{ I quadrant}$
${\theta}_{2} = {\tan}^{-} 1 \left(- \frac{7}{5}\right) \approx {305.54}^{\circ} , \text{ IV quadrant}$

${z}_{1} + {z}_{2} = \sqrt{85} \cos \left(77.47\right) + \sqrt{74} \cos \left(305.54\right) + i \left(\sqrt{85} \sin 77.47 + \sqrt{74} \sin 305.54\right)$

$\implies 2 + 5 + i \left(9 - 7\right)$

color(violet)(=> 7 + 2 i