Blog: `1 + i` Write the trigonometric form of the number.

Friday, August 10, 2012

$\displaystyle{1}+{i}$ Write the trigonometric form of the number.

To convert a complex number

to its trigonometric form

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

the formulas needed are:

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

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

Applying these two formulas, the r and $\displaystyle\theta$ of

$\displaystyle{z}={1}+{i}$

will be:

$\displaystyle{r}=\sqrt{{{{1}}^{{2}}+{{1}}^{{2}}}}=\sqrt{{2}}$

$\displaystyle\theta={{\tan}}^{{-{1}}}{\left(\frac{{1}}{{1}}\right)}={{\tan}}^{{-{1}}}{1}={{45}}^{{o}}$

Plugging them to

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

result to:

$\displaystyle{z}=\sqrt{{2}}{\left({{\cos{{45}}}}^{{o}}+{i}{{\sin{{45}}}}^{{o}}\right)}$

Therefore, the trigonometric form of $\displaystyle{1}+{i}$ is $\displaystyle\sqrt{{2}}{\left({{\cos{{45}}}}^{\oplus}{i}{{\sin{{45}}}}^{{o}}\right)}$ .

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Friday, August 10, 2012

$\displaystyle{1}+{i}$ Write the trigonometric form of the number.

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