Blog: `8 + 3i` Write the trigonometric form of the number.

Monday, November 14, 2016

$\displaystyle{8}+{3}{i}$ Write the trigonometric form of the number.

$\displaystyle{8}+{3}{i}$

The trigonometric form of a complex number $\displaystyle{z}={x}+{y}{i}$ is

$\displaystyle{z}={r}{\left({\cos{\theta}}+{i}{\sin{\theta}}\right)}$

where

$\displaystyle{r}=\sqrt{{{{x}}^{{2}}+{{y}}^{{2}}}}$

$\displaystyle\theta={{\tan}}^{{-{1}}}\frac{{y}}{{x}}$

Applying these formulas, the values of r and theta of z=8+3i are:

$\displaystyle{r}=\sqrt{{{{8}}^{{2}}+{{3}}^{{2}}}}=\sqrt{{{64}+{9}}}=\sqrt{{73}}$

$\displaystyle\theta={{\tan}}^{{-{1}}}\frac{{3}}{{8}}={20.56}$

Plug-in them to the trigonometric form.

$\displaystyle{z}=\sqrt{{73}}{\left({{\cos{{20.56}}}}^{\oplus}{i}{{\sin{{20.56}}}}^{{o}}\right)}$

Thus, the trigonometric form of $\displaystyle{8}+{3}{i}$ is $\displaystyle\sqrt{{73}}{\left({{\cos{{20.56}}}}^{{o}}+{i}{{\sin{{20.56}}}}^{{o}}\right)}$ .

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Monday, November 14, 2016

$\displaystyle{8}+{3}{i}$ Write the trigonometric form of the number.

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