Question

# Write the complex number in trigonometric form r(cos theta + i sin theta),with theta in the interval [0^@, 360^@]5sqrt3 + 5i

Complex numbers

Write the complex number in trigonometric form $$r(\cos \theta + i \sin \theta)$$,with theta in the interval $$\displaystyle{\left[{0}^{\circ},{360}^{\circ}\right]}$$
$$\displaystyle{5}\sqrt{{3}}+{5}{i}$$

2021-02-16
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 $$\displaystyle{10}{\left({\cos{{\left(\frac{\pi}{{6}}\right)}}}+{i}{\sin{{\left(\frac{\pi}{{6}}\right)}}}\right.}$$