Blog: `-7 + 4i` Write the trigonometric form of the number.

Monday, June 20, 2011

$\displaystyle-{7}+{4}{i}$ Write the trigonometric form of the number.

$\displaystyle-{7}+{4}{i}$

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 formulas, the values of r and theta pf x=-7+4i are:

$\displaystyle{r}=\sqrt{{{{\left(-{7}\right)}}^{{2}}+{{4}}^{{2}}}}=\sqrt{{{49}+{16}}}=\sqrt{{65}}$

$\displaystyle\theta={{\tan}}^{{-{1}}}\frac{{-{7}}}{{4}}=-{{29.744813}}^{{o}}$

Since x is negative and y is positive, the angle is located at the second quadrant. The equivalent positive angle of theta is:

$\displaystyle\theta={{180}}^{{o}}+{\left(-{{29.744813}}^{{o}}\right)}={{150.2551870}}^{{o}}$

Rounding off to two decimal places, it becomes:

$\displaystyle\theta={{150.26}}^{{o}}$

Plugging the values of r and theta to the trigonometric form yields:

$\displaystyle{z}=\sqrt{{65}}{\left({{\cos{{150.26}}}}^{{o}}+{i}{{\sin{{150.26}}}}^{{o}}\right)}$

Therefore, the trigonometric form of $\displaystyle-{7}+{4}{i}$ is $\displaystyle\sqrt{{65}}{\left({{\cos{{150.26}}}}^{{o}}+{i}{{\sin{{150.26}}}}^{{o}}\right)}$ .

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Monday, June 20, 2011

$\displaystyle-{7}+{4}{i}$ Write the trigonometric form of the number.

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