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

Question
Complex numbers
asked 2021-02-15
Write the complex number in trigonometric form r(cos theta + i sin theta),with theta in the interval \(\displaystyle{\left[{0}^{\circ},{360}^{\circ}\right]}\)
\(\displaystyle{5}\sqrt{{3}}+{5}{i}\)

Answers (1)

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 \(\displaystyle{10}{\left({\cos{{\left(\frac{\pi}{{6}}\right)}}}+{i}{\sin{{\left(\frac{\pi}{{6}}\right)}}}\right.}\)
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