Blog: `5 + 2i` Write the trigonometric form of the number.

Sunday, November 2, 2008

$\displaystyle{5}+{2}{i}$ Write the trigonometric form of the number.

$\displaystyle{5}+{2}{i}$

Take note that the trigonometric form of a complex number z=x+yi is

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

where

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

and

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

Applying these formula, the values of r and theta of $\displaystyle{z}={5}+{2}{i}$ are:

$\displaystyle{r}=\sqrt{{{{5}}^{{2}}+{{2}}^{{2}}}}=\sqrt{{{25}+{4}}}=\sqrt{{29}}$

$\displaystyle\theta={{\tan}}^{{-{1}}}\frac{{2}}{{5}}={{21.8}}^{{o}}$

Plugging them to the trigonometric form, it result to:

$\displaystyle{z}=\sqrt{{29}}{\left({{\cos{{21.8}}}}^{{o}}+{i}{{\sin{{21.8}}}}^{{o}}\right)}$

Thus, the trigonometric form of $\displaystyle{5}+{2}{i}$ is $\displaystyle\sqrt{{29}}{\left({{\cos{{21.8}}}}^{{o}}+{i}{{\sin{{21.8}}}}^{{o}}\right)}$ .

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Sunday, November 2, 2008

$\displaystyle{5}+{2}{i}$ Write the trigonometric form of the number.

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