The trigonometric form of the complex number is \(\displaystyle{r}{\left({\cos{\theta}}+{i}{\sin{\theta}}\right)}.\)

Here, r is the modulus of the complex number and theta is called the argument of the complex number.

The formula for the modulus of any complex number a+bi is defined as, \(\displaystyle{r}=\sqrt{{{a}^{{2}}+{b}^{{2}}}}\)

Substitute \(\displaystyle{a}={5}\sqrt{{3}},{b}={5}\) in the formula for modulus.

\(\displaystyle{r}=\sqrt{{{\left({5}\sqrt{{3}}\right)}^{{2}}+{5}^{{2}}=\sqrt{{{75}+{25}}}=\sqrt{{100}}={10}}}\)

Since r is the modulus, it is always positive. So, r= 10.

The formula for the argument of the complex number is defined as, \(\displaystyle\theta={{\tan}^{{-{{1}}}}{\left(\frac{{b}}{{a}}\right)}}\)

Substitute \(\displaystyle{a}={5}\sqrt{{3}},{b}={5}\) in the formula for argument

\(\displaystyle\theta={{\tan}^{{-{{1}}}}{\left(\frac{{5}}{{{5}\sqrt{{3}}}}\right)}}={{\tan}^{{-{{1}}}}{\left(\frac{{1}}{\sqrt{{3}}}\right)}}\)

The value of tan is \(\displaystyle\frac{{1}}{\sqrt{{3}}}\) for \(\displaystyle\frac{\pi}{{6}}\)

So, \(\displaystyle\theta=\frac{\pi}{{6}}\)

So, the trigonometric form of the given complex number is \(\displaystyle{10}{\left({\cos{{\left(\frac{\pi}{{6}}\right)}}}+{i}{\sin{{\left(\frac{\pi}{{6}}\right)}}}\right.}\)

Here, r is the modulus of the complex number and theta is called the argument of the complex number.

The formula for the modulus of any complex number a+bi is defined as, \(\displaystyle{r}=\sqrt{{{a}^{{2}}+{b}^{{2}}}}\)

Substitute \(\displaystyle{a}={5}\sqrt{{3}},{b}={5}\) in the formula for modulus.

\(\displaystyle{r}=\sqrt{{{\left({5}\sqrt{{3}}\right)}^{{2}}+{5}^{{2}}=\sqrt{{{75}+{25}}}=\sqrt{{100}}={10}}}\)

Since r is the modulus, it is always positive. So, r= 10.

The formula for the argument of the complex number is defined as, \(\displaystyle\theta={{\tan}^{{-{{1}}}}{\left(\frac{{b}}{{a}}\right)}}\)

Substitute \(\displaystyle{a}={5}\sqrt{{3}},{b}={5}\) in the formula for argument

\(\displaystyle\theta={{\tan}^{{-{{1}}}}{\left(\frac{{5}}{{{5}\sqrt{{3}}}}\right)}}={{\tan}^{{-{{1}}}}{\left(\frac{{1}}{\sqrt{{3}}}\right)}}\)

The value of tan is \(\displaystyle\frac{{1}}{\sqrt{{3}}}\) for \(\displaystyle\frac{\pi}{{6}}\)

So, \(\displaystyle\theta=\frac{\pi}{{6}}\)

So, the trigonometric form of the given complex number is \(\displaystyle{10}{\left({\cos{{\left(\frac{\pi}{{6}}\right)}}}+{i}{\sin{{\left(\frac{\pi}{{6}}\right)}}}\right.}\)